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Annals of Combinatorics

, Volume 23, Issue 3–4, pp 1039–1072 | Cite as

Singular Overpartitions and Partitions with Prescribed Hook Differences

  • Seunghyun Seo
  • Ae Ja YeeEmail author
Article
  • 15 Downloads

Abstract

Singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row, were first introduced by Andrews in 2015. In his paper, Andrews investigated an interesting subclass of singular overpartitions, namely, (Ki)-singular overpartitions for integers Ki with \( 1\le i<K/2\). The definition of such singular overpartitions requires successive ranks, parity blocks and anchors. The concept of successive ranks was extensively generalized to hook differences by Andrews, Baxter, Bressoud, Burge, Forrester and Viennot in 1987. In this paper, employing hook differences, we generalize parity blocks. Using this combinatorial concept, we define \((K,i,\alpha , \beta )\)-singular overpartitions for positive integers \(\alpha , \beta \) with \(\alpha +\beta <K\), and then we show some connections between such singular overpartitions and ordinary partitions.

Keywords

Partitions Overpartitions Singular overpartitions Frobenius symbols Successive ranks Hook differences 

Mathematics Subject Classification

Primary 05A17 Secondary 11P81 

1 Introduction

A partition of a positive integer n is a weakly decreasing sequence of positive integers whose sum equals n [3]. The integers in the sequence are called parts. An overpartition of n is a partition in which the first occurrence of a part may be overlined [12].

A Frobenius symbol is a two-rowed array of nonnegative integers such that entries in each row are strictly decreasing and the numbers of entries in the top and bottom rows are equal [4]. There is a natural one-to-one correspondence between partitions and Frobenius symbols [4, 15, 18]. For self-containedness, the definition of the Frobenius symbol of a partition is given later in Definition 2.8. For an overpartition, one can define the corresponding Frobenius symbol by allowing overlined entries in a similar way.

In [5], Andrews introduced singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row. For integers Ki with \(1\le i<K/2\), Andrews defined a subclass of singular overpartitions with some restrictions subject to K and i, namely (Ki)-singular overpartitions. He then showed interesting combinatorial and arithmetic properties of (Ki)-singular overpartitions. As seen in [5], (Ki)-singular overpartitions are closely related to partitions counted by partition sieves, which were first employed by Andrews [1, 2] to discover Rogers–Ramanujan type partitions and later generalized further by Bressoud [7].

Successive ranks are the differences between the top and bottom entries of the columns in a Frobenius symbol. In the partition sieves, they are vital combinatorial statistics and have led to a number of discoveries of Rogers–Ramanujan type partitions [1, 2, 7, 9, 10, 11]. The concept of successive ranks was extensively generalized to hook differences by Andrews, Baxter, Bressoud, Burge, Forrester and Viennot in [6], which concerns partitions with prescribed hook differences. The work in [6] was further extended by Gessel and Krattenthaler [14].

The main purpose of this paper is to generalize (Ki)-singular overpartitions by utilizing the concept of hook differences. Throughout this paper, we assume that K, i, \(\alpha \) and \(\beta \) are positive integers with \(i<K/2\) and \(\alpha +\beta <K\). For a positive integer n, let \({\overline{Q}}_{K, i, \alpha , \beta }(n)\) be the number of singular overpartitions of n with prescribed overlining constraints subject to Ki, \(\alpha \) and \(\beta \). Such singular overpartitions will be called \((K,i,\alpha ,\beta )\)-singular overpartitions. Because of the complexity of the constraints, we defer the exact definition of \((K,i,\alpha ,\beta )\)-singular overpartitions to Sect. 2.

One of our results is given in the following theorem.

Theorem 1.1

We have
$$\begin{aligned} \sum _{n=0}^{\infty } {\overline{Q}}_{K,i,\alpha , \alpha }(n) q^n&=\frac{(-q^{i\alpha };q^{K\alpha })_{\infty } (-q^{(K-i)\alpha };q^{K\alpha })_{\infty } (q^{K\alpha };q^{K\alpha })_{\infty }}{(q;q)_{\infty }}, \end{aligned}$$
where \({\overline{Q}}_{K,i,\alpha , \alpha }(0)=1\) and \((a;q)_{\infty }=\lim _{n\rightarrow \infty } \prod _{j=0}^n (1-aq^j)\).

We note that \({\overline{Q}}_{K,i, 1, 1}(n)\) becomes the number of (Ki)-singular overpartitions of n given by Andrews in [5].

For any positive integers m and n, let us define a refined partition function \({\overline{Q}}_{K,i,\alpha , \beta }(m,n)\) by the number of \((K,i,\alpha , \beta )\)-singular overpartitions of n with an overlined entry in its mth anchor. Again, because of the complexity, anchors are defined in Sect. 2. Then we have the following theorem.

Theorem 1.2

For \(m\ge 1\) and \(n\ge 0\),
$$\begin{aligned} {\overline{Q}}_{K,i, \alpha , \alpha } (m, n)=p\bigg (n-\alpha K\left( {\begin{array}{c}m+1\\ 2\end{array}}\right) +\alpha i m \bigg )+p\bigg (n-\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) -\alpha i m\bigg ), \end{aligned}$$
where p(N) denotes the number of ordinary partitions of N with \(p(0)=1\) and \(p(N)=0\) for \(N<0\).

For arbitrary positive integers \(\alpha , \beta \), more general and refined results than Theorems 1.1 and 1.2 are presented in Sect. 6. Our proofs are combinatorial and bijective generalizing the proof methods used in [15]. One of the main ingredients of the methods in [15] was Dyson’s map [13]. We will generalize this map for our purpose.

The rest of this paper is organized as follows. In Sect. 2, some basic definitions and notions are recollected followed by the definition of \((K,i,\alpha ,\beta )\)-singular overpartitions. In Sect. 3, Dyson’s map and its generalization are presented along with the shift map from [15]. Necessary lemmas for later use are given in Sect. 4. In Sect. 5, another representation of \((K,i,\alpha ,\beta )\)-singular overpartitions is given and it is shown bijectively that \((K,i,\alpha ,\beta )\)-singular overpartitions are related to ordinary partitions. In Sect. 6, our theorem on \((K,i,\alpha ,\beta )\)-singular overpartitions is proved along with Theorems 1.1 and 1.2. Some remarks are given in Sect. 7.

2 \((K,i,\alpha ,\beta )\)-Singular Overpartitions

For a partition \(\lambda \) of n, we denote it by \(\lambda \vdash n\), the sum of parts by \(|\lambda |\), and the number of parts by \(\ell (\lambda )\). The Ferrers diagram of \(\lambda \) is a left-justified graphical representation whose jth row has as many boxes as the jth part \(\lambda _j\). The box in row x and column y of the Ferrers diagram is called node (xy). If a node (xy) is inside the Ferrers diagram, i.e., \(1\le x\le \ell (\lambda )\) and \(1\le y \le \lambda _{x}\), then we denote it by \((x,y)\in \lambda \). For an integer k, the diagonal diag \(=k\) is the line passing through nodes (xy) with \(x=y+k\) [6]. Figure 1 shows some diagonals on the partition (5, 4, 2, 2).
Fig. 1

\(\lambda =(5,4,2,2)\) with diagonals

The conjugate of \(\lambda \) is the partition resulting from reflecting the Ferrers diagram of \(\lambda \) about the main diagonal, and we denote the conjugate partition by \(\lambda '\). For instance, \(\lambda '=(4,4,2,2,1)\) in Fig. 1.

Definition 2.1

For a partition \(\lambda \), we define the hook difference at a node \((x,y) \in \lambda \) as
$$\begin{aligned} h_{(x,y)}=h_{(x,y)}(\lambda )&=(\lambda _x-y)-(\lambda '_y-x)=\lambda _x-\lambda '_y+(x-y). \end{aligned}$$
(2.1)
For convenience, we also define the hook difference at a node \((x,y)\not \in \lambda \) as
$$\begin{aligned} h_{(x,y)}=h_{(x,y)}(\lambda )&={\left\{ \begin{array}{ll} -\infty , &{} \hbox { if}\ x>y,\\ 0, &{} \hbox { if}\ x=y, \\ +\infty , &{} \hbox { if}\ x<y. \end{array}\right. }\qquad \qquad \qquad \qquad \quad \;\; \end{aligned}$$
(2.2)
Here we note that the hook difference at a node \((x,y)\in \lambda \) is defined as \(\lambda _x -\lambda '_y\) in [6].
Fig. 2

\(\lambda =(5,4,2,2)\) and its hook at (1, 2)

Figure 2 shows the hook at node (1, 2) in the partition (5, 4, 2, 2) and its hook difference equals 0. For a node not in \(\lambda \), (2.2) says that the hook difference at that node is defined to be 0 if the node is on the main diagonal, \(-\infty \) if it is below the main diagonal, and \(\infty \) if it is above the main diagonal.

In the next two lemmas, we will show how the hook difference at a node on the diagonal \(1-\beta \) affects the hook differences at nodes on the diagonal \(\alpha -1\), and vice versa.

Lemma 2.2

For a partition \(\lambda \), suppose that \(h_{(j,j+\beta -1)}(\lambda ) \le 1-i\). Then, for any nonnegative integer x with \(0\le x \le \beta -1\),
$$\begin{aligned} h_{(j+x+\alpha -1, j+x)} \le K-i-2. \end{aligned}$$

Proof

Let us consider two cases: \((j,j+\beta -1)\in \lambda \) and \((j,j+\beta -1)\notin \lambda \).
  • Case 1: \((j,j+\beta -1)\in \lambda \). If \((j+x+\alpha -1, j+x) \in \lambda \), then
    $$\begin{aligned} h_{(j+x+\alpha -1, j+x)}&=\lambda _{j+x+\alpha -1}-\lambda '_{j+x} +\alpha -1 \\&\le \lambda _{j}-\lambda '_{j+x} +\alpha -1\\&\le \lambda _j-\lambda '_{j+\beta -1}+\alpha -1 \\&=h_{(j,j+\beta -1)} + (\alpha +\beta -2)\\&\le 1-i + (\alpha + \beta )-2\\&\le K-i-2, \end{aligned}$$
    where the second to last inequality follows from \(h_{(j, j+\beta -1)} \le 1-i\) and the last inequality follows from \(K>\alpha +\beta \). If \((j+x+\alpha -1, j+x) \notin \lambda \), then
    $$\begin{aligned} h_{(j+x+\alpha -1, j+x)} \le 0 \le K-i-2, \end{aligned}$$
    where the left inequality follows from (2.2) and the right inequality follows from \(1\le i <K/2\).
  • Case 2: \((j,j+\beta -1)\notin \lambda \). Then, by (2.2), we know that
    $$\begin{aligned} h_{(j,j+\beta -1)}=0~~ \text {or}~~+\infty . \end{aligned}$$
    However, since \(h_{(j,j+\beta -1)} \le 1-i\), it has to be
    $$\begin{aligned} h_{(j,j+\beta -1)} =0, \end{aligned}$$
    and then, by (2.2), it has to be \(\beta =1\), i.e., \((j,j) \notin \lambda \). Then, for any nonnegative integer x, \((j+x+\alpha -1, j+x) \notin \lambda \). As seen above in the second case in Case 1, we get the desired inequality.
\(\square \)
Lemma 2.2 can be summarized as follows. In Fig. 3, if \(h_{a}\le 1-i\), then \(h_{z}\le K-i-2\) for any nodes z between c and d. Further explanations on nodes (jj) and \((j',j')\) will be given in Remark 2.7.
Fig. 3

Lemma 2.2

Lemma 2.3

For a partition \(\lambda \), suppose that \(h_{(j+\alpha -1, j)}(\lambda ) \ge K-i-1\). Then, for any nonnegative integer x with \(0 \le x \le \alpha -1\),
$$\begin{aligned} h_{(j+x, j+x+\beta -1)}(\lambda ) \ge 2-i . \end{aligned}$$

Proof

By conjugation, it is clear that
$$\begin{aligned} (j+\alpha -1,j)\in \lambda \quad \text {if and only if} \quad (j, j+\alpha -1)\in \lambda ' \end{aligned}$$
and
$$\begin{aligned} h_{(j+\alpha -1,j)}(\lambda )=-h_{(j, j+\alpha -1)}(\lambda '). \end{aligned}$$
Thus, by the assumption that \(h_{(j+\alpha -1, j)}(\lambda ) \ge K-i-1\), we have
$$\begin{aligned} h_{(j,j+\alpha -1)}(\lambda ') \le 1-(K-i), \end{aligned}$$
which implies by Lemma 2.2 that
$$\begin{aligned} h_{(j+x+\beta -1,j+x)}(\lambda ') \le K-(K-i)-2. \end{aligned}$$
This is equivalent to
$$\begin{aligned} h_{(j+x,j+x+\beta -1)}(\lambda ) \ge 2-i, \end{aligned}$$
as desired. \(\square \)
Lemma 2.3 can be summarized as follows. In Fig. 4, if \(h_{c}\ge K-i-1\), then \(h_{z}\ge 2-i\) for any nodes z between a and b. Further explanations on nodes (jj) and \((j',j')\) will be given in Remark 2.7.
Fig. 4

Lemma 2.3

Remark 2.4

By the cases when \(x=0\) in Lemmas 2.2 and 2.3, we see that it does not happen simultaneously that \(h_{(j, j+\beta -1)} \le 1-i\) and \(h_{(j+\alpha -1, j)} \ge K-i-1\) for any j. That is, in Fig. 3 or 4, it is impossible that \(h_{a}\le 1-i\) and \(h_{c}\ge K-i-1\) hold at the same time.

We now define the sign of a node on the main diagonal. For a node (jj), its sign will be determined by the hook differences at the nodes that are in the hook of (jj) and on the diagonals \(1-\beta \) or \(\alpha -1\).

Definition 2.5

Let \(\lambda \) be a partition. For a positive integer j, suppose that a node \((j,j) \in \lambda \). Then the node (jj) is said to be
  • \((K,i, \alpha , \beta )\)-negative if
    $$\begin{aligned} (j,j+\beta -1)\in \lambda \quad \text {and}\quad h_{(j,j+\beta -1)} \le 1-i; \end{aligned}$$
    (2.3)
  • \((K,i, \alpha , \beta )\)-positive if
    $$\begin{aligned} (j+\alpha -1, j) \in \lambda \quad \text {and}\quad h_{(j+\alpha -1, j)} \ge K-i -1; \end{aligned}$$
    (2.4)
  • \((K,i, \alpha , \beta )\)-neutral otherwise.

By Remark 2.4, we see that the node (jj) cannot be \((K,i,\alpha , \beta )\)-negative and positive at the same time. Also, we see that the node (jj) is \((K,i,\alpha , \beta )\)-neutral if and only if
$$\begin{aligned} h_{(j,j+\beta -1)} \ge 2-i \quad \text {and} \quad h_{(j+\alpha -1,j)}\le K-i-2, \end{aligned}$$
which works even if \((j,j+\beta -1)\not \in \lambda \) or \((j+\beta -1,j) \not \in \lambda \) due to (2.2).
In Fig. 5, we set \((K,i,\alpha , \beta )=(7,2,3,2)\). The number in each box on the diagonals \(-1\) and 2 is the hook difference at the node. Then the sign of node (1, 1) is positive since \(h_{(1,2)}=3>1-i=-1\), but \( h_{(3,1)}=5\ge K-i-1=4\). The signs of the other nodes on the main diagonal can be determined in the same way. Here we note that the sign of node (5, 5) is neutral since there are no nodes in the hook of (5, 5) that are on the diagonals \(-1\) or 2. The letters p, n and e in the boxes stand for positive, negative and neutral, respectively.
Fig. 5

(7, 2, 3, 2)-positive, negative and neutral

Remark 2.6

By the definition, if a node (jj) is \((K,i,\alpha , \beta )\)-negative, then \((j,j+\beta -1)\in \lambda \), so \(\lambda _{j}\ge j+\beta -1\). Similarly, if a node (jj) is \((K,i,\alpha , \beta )\)-positive, then \((j+\alpha -1, j)\in \lambda \), so \(\lambda '_j\ge j+\alpha -1\).

Remark 2.7

By Lemmas 2.2 and 2.3, we also see that the following statements hold true:
  1. (i)

    Suppose that a node (jj) is \((K,i,\alpha ,\beta )\)-negative. Then for any x with \(1\le x\le \beta -1\) and \((j+x,j+x)\in \lambda \), a node \((j+x,j+x)\) cannot be \((K,i,\alpha ,\beta )\)-positive. That is, in Fig. 3, if (jj) is negative, then any nodes between (jj) and \((j',j')\) are negative or neutral.

     
  2. (ii)

    Suppose that a node (jj) is \((K,i,\alpha ,\beta )\)-positive. Then for any x with \(1\le x\le \alpha -1\) and \((j+x, j+x)\in \lambda \), a node \((j+x, j+x)\) cannot be \((K,i,\alpha ,\beta )\)-negative. That is, in Fig. 4, if (jj) is positive, then any nodes between (jj) and \((j',j')\) are positive or neutral.

     

Next, we give the definition of Frobenius symbols and then rephrase our definition of the sign of nodes in terms of Frobenius symbol.

Definition 2.8

For a partition \(\lambda \), let \(\delta \) be the largest x such that \(\lambda _x\ge x\). Then the Frobenius symbol of \(\lambda \) is defined as
$$\begin{aligned} \lambda =\left( \begin{array}{lll} a_1 &{} \cdots &{} a_{\delta }\\ b_1&{} \cdots &{} b_{\delta } \end{array} \right) , \end{aligned}$$
where \(a_x=\lambda _x-x\) and \(b_y=\lambda '_y-y\) for \(1\le x, y\le \delta \).
For \(1\le j\le \delta \), the jth column of the Frobenius symbol of \(\lambda \) is said to be
  • \((K,i,\alpha ,\beta )\)-negative if the node \((j,j)\in \lambda \) is \((K,i,\alpha ,\beta )\)-negative;

  • \((K,i,\alpha ,\beta )\)-positive if the node \((j,j)\in \lambda \) is \((K,i,\alpha ,\beta )\)-positive;

  • \((K,i,\alpha ,\beta )\)-neutral if the node \((j,j)\in \lambda \) is \((K,i,\alpha ,\beta )\)-neutral.

For instance, the partition in Fig. 5 can be written as follows:
$$\begin{aligned} \left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 11&{} 9 &{}8 &{} 2 &{} 0 \\ 7 &{} 6 &{} 4&{} 2&{} 1 \end{array} \right) , \end{aligned}$$
and the first column is positive, the fourth column is negative, and the others are neutral.
We note that from the definitions of hook differences and Frobenius symbols, it is clear that for \(1\le x,y \le \delta \),
$$\begin{aligned} h_{(x,y)}=a_x-b_y +2(x-y). \end{aligned}$$
(2.5)

Remark 2.9

When we compute hook differences from a Frobenius symbol, perhaps it is more convenient to write the Frobenius symbol in a slightly different form by shifting the second row to the right or left by \(\alpha -1\) units or \(\beta -1\) units.

In particular, let \(\delta \) be the number of columns in the Frobenius symbol and \( \delta \ge \max (\alpha , \beta )\). Then we can explicitly write the conditions for a node being \((K,i,\alpha ,\beta )\)-negative, positive, or neutral in terms of the entries in the Frobenius symbol. If \(\delta <\min (\alpha , \beta )\), then it is better to use Definitions 2.5 and 2.8 with (2.1).

In what follows, assuming that \( \delta \ge \max (\alpha , \beta )\), we explain how to determine if a node is \((K,i,\alpha ,\beta )\)-negative or positive.
  1. (i)
    To check more easily if a node is \((K,i,\alpha ,\beta )\)-negative, we write the Frobenius symbol as follows:
    $$\begin{aligned} \begin{array}{lllllll} &{} &{} a_1 &{} \cdots &{} a_{\delta -\beta +1} &{} \cdots &{} a_{\delta }\\ b_1&{} \cdots &{} b_{\beta }&{} \cdots &{} b_{\delta }. \end{array} \end{aligned}$$
    Then, the hook differences to be needed to compute are the difference of the entries in each column with two entries. Namely, for \(1\le j\le \delta -\beta +1\), by (2.3) and (2.5), if
    $$\begin{aligned} a_{j}-b_{\beta +j-1} \le 2\beta -1-i, \end{aligned}$$
    (2.6)
    then the node (jj) is negative. Let us turn to nodes (jj) for \(\delta -\beta +2\le j\le \delta \). See the figure below. If the node p was negative, i.e., (2.6) holds true for \(j=\delta -\beta +1\), then next \(\beta -1\) nodes on the main diagonal after p could not be positive by Remark 2.7 (i). Thus, the nodes (jj) between p and \(p'\) would be negative or neutral. However, if the node p was not negative, then the next node might be positive.
     

To compute the hook difference, the Ferrers diagram is more convenient to use than the Frobenius symbol.

  1. (ii)
    Similarly, to check if a node is \((K,i,\alpha ,\beta )\)-positive, we write the Frobenius symbol as follows.
    $$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} a_1 &{} \cdots &{} a_{\alpha } &{} \cdots &{} a_{\delta }\\ &{} &{} b_1&{} \cdots &{} b_{\delta -\alpha +1}&{} \cdots &{} b_{\delta } \end{array} \end{aligned}$$
    Then, the hook differences to be needed to compute are the difference of the entries in each column with two entries. For \(1\le j\le \delta -\alpha +1\), by (2.4) and (2.5), if
    $$\begin{aligned} a_{\alpha +j-1}-b_j\ge K-i+1-2\alpha , \end{aligned}$$
    (2.7)
    then the node (jj) is positive. Similarly, we can show that nodes (jj) for \(\delta -\alpha +2\le j\le \delta \) cannot be positive if the node \((\delta -\alpha +1,\delta -\alpha +1)\) is positive, i.e., (2.7) holds true for \(j=\delta -\alpha +1\).
     
  2. (iii)
    Adopting the notation for cylindric partitions [14], we may combine the expressions in (i) and (ii) above as follows:
    $$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} &{} &{} &{} &{} a_1 &{} \cdots &{} \cdots &{} \cdots &{} a_{\delta }\\ &{} &{} b_1&{} \cdots &{} b_{\beta }&{} \cdots &{} \cdots &{} b_{\delta } \\ a_1 &{} \cdots &{} a_{\alpha } &{} \cdots &{} \cdots &{} a_{\delta }. \end{array} \end{aligned}$$
    We note that it should be checked if the difference between the first and second rows satisfies (2.6) and the difference between the second and third rows satisfies (2.7).
     
We give an example to demonstrate how to find positive or negative columns in a Frobenius symbol.

Example 2.10

Consider the following Frobenius symbol:
$$\begin{aligned} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 31 &{} 28 &{} 27&{} 22&{} 18 &{} 9 &{} 8 &{} 7&{} 1&{} 0\\ 29 &{} 26 &{} 25&{} 23 &{} 22 &{} 8&{} 5&{} 4&{} 1 &{} 0 \end{array} \right) . \end{aligned}$$
Let \((K,i,\alpha , \beta )=(5,2,2,2)\).
First, for negative nodes, by (2.6), we compute and see if
$$\begin{aligned} a_{j}-b_{j+1} \le 1. \end{aligned}$$
So, we shift the second row to the left by 1 unit and compute the difference of the entries in each column with two entries. Again, the upper entries with difference at most 1 are circled:These circled entries indicate the negative columns in the original Frobenius symbol. Since \(\beta =2\) and the second to last column is negative, the last column cannot be positive. Actually, from the Ferrers diagram of the Frobenius symbol, we see that the node (10, 11) does not exist in the partition, so the sign of the last column is neutral.
Similarly, for positive nodes, by (2.7), we compute and see if
$$\begin{aligned} a_{j+1}-b_{j} \ge 0. \end{aligned}$$
So, we shift the first row of the Frobenius symbol to the left by 1 unit and compute the difference of the entries in each column with two entries. For convenience, if the difference is at least 0, then the lower entry will be circled:These circled entries indicate the positive columns in the original Frobenius symbol. For the last column, we cannot say if it is not positive because the second to last column is not positive. However from the earlier calculations for negative nodes, we already know that it is neutral.
We now take the first row in (2.8) and the second row in (2.9) to form the following Frobenius symbol:where columns with circled entries in the first row are negative and columns with circled entries in the second row are positive. Therefore, we have
$$\begin{aligned} \left( \begin{array}{cccccccccc} &{} p &{} &{} n&{} &{} p &{} p &{} &{}n &{} \\ 31 &{} 28 &{} 27&{} 22 &{} 18 &{} 9 &{} 8 &{} 7&{} 1 &{} 0\\ 29 &{} 26 &{} 25&{} 23 &{} 22 &{} 8 &{} 5 &{} 4&{} 1 &{} 0 \end{array} \right) , \end{aligned}$$
where p and n indicate that the corresponding columns are positive and negative, respectively.

2.1 \((K,i,\alpha , \beta )\)-Parity Blocks and Anchors

Following [5], we generalize the notion of parity blocks of Frobenius symbols. Unlike in [5], for convenience, we introduce neutral blocks. For a Frobenius symbol, if there are consecutive neutral columns starting from the first column, then separate them to form a neutral block. We shall say that the block is neutral and we denote it by E. For the remaining columns, we take sets of contiguous columns maximally extended to the right, where all the columns have either the same parity or neutral. We shall say that a block is positive (or negative) if it contains no negative (or no positive, resp.) nodes, and we denote it by P (or N, resp.).

Example 2.11

We consider the same Frobenius symbol as in Example 2.10. Then, the (5, 2, 2, 2)-parity blocks are
$$\begin{aligned} \left( \begin{array}{c|c|c|c|c} \overset{E}{31} &{} \overset{P}{28\;\; 27 } &{} \overset{N}{22 \;\; 18} &{} \overset{P}{9 \;\; 8 \;\; 7 } &{} \overset{N}{ 1\;\; 0}\\ 29 &{} 26 \;\; 25 &{} 23 \;\; 22 &{} 8\;\; 5\;\; 4&{} 1 \;\; 0 \end{array} \right) , \end{aligned}$$
where the superscripts E, P, and N denote that the block is neutral, positive, and negative, respectively.

Definition 2.12

For a positive (or negative) block, we define its anchor as the first column in the block.

Lemma 2.13

Let \(\lambda \) be a partition. For a non-last block of \(\lambda \), if it is positive or negative, then there are at least \(\alpha \) or \(\beta \) columns, respectively.

This lemma follows from Remark 2.7 on the sign of \(\alpha -1\) (or \(\beta -1\)) nodes next to a positive (or negative, resp.) node. So we omit the proof.

Example 2.14

Let \((K,i,\alpha , \beta )=(7,3,3,2)\). We consider the following Frobenius symbol:
$$\begin{aligned} \left( \begin{array}{ccccccccccc} 31 &{} 28 &{} 27&{} 22&{} 18 &{} 10&{} 9 &{} 8 &{} 7&{} 1\\ 29 &{} 26 &{} 25&{} 23 &{} 22 &{}20&{} 8&{} 5&{} 4&{} 3 \end{array} \right) . \end{aligned}$$
Following Example 2.10, we will decompose this Frobenius symbol into (7, 3, 3, 2)-parity blocks and identify their anchors.
First, by (2.6), for negative nodes, we compute and see if
$$\begin{aligned} a_{j}-b_{j+1} \le 0. \end{aligned}$$
So, we shift the second row to the left by 1 unit and compute the difference of the entries in each column with two entries to see if the difference is at most 0.where the circled 22, 18 and 1 indicate the negative columns in the original Frobenius symbol. Here, we note that since the second to last column is not negative, we cannot determine the sign of the last column from the sign of the second to last column, so we used the Ferrers diagram to compute the hook difference at the corresponding node (10, 11), which is 0.
Similarly, by (2.7), for positive nodes, we compute and see if
$$\begin{aligned} a_{j+2}-b_{j} \ge -1. \end{aligned}$$
We shift the first row of the Frobenius symbol to the left by 2 units and compute the difference of the entries in each column with two entries.where the circled 8 indicates the positive column in the original Frobenius symbol. Here, we used the Ferrers diagram to determine the sign of each of the last two columns.
Therefore, we see that the (7, 3, 3, 2)-parity blocks with anchors are as follows:
$$\begin{aligned} \left( \begin{array}{c|c|c|c} \overset{E}{31 \;\; 28 \;\; {27} }&{} \overset{N}{ \; {22} \;\; 18 \;\; 10} &{} \overset{P}{ \boxed {9} \; 8 \;\; {7}} &{} \overset{N}{ { 1} } \\ 29 \;\; 26 \;\; 25&{} \boxed {23} \; 22 \;\; 20 &{} \;\; {8} \;\; 5\;\; 4\; &{} \boxed {3} \end{array} \right) , \end{aligned}$$
where the bottom entries in the anchors of the negative blocks are boxed and the top entry in the anchor of the positive block is boxed.

2.2 \((K,i,\alpha ,\beta )\)-Singular Overpartitions

We are now ready to define \((K,i, \alpha ,\beta )\)-singular overpartitions. An overpartition is \((K, i, \alpha ,\beta )\)-singular if its Frobenius symbol satisfies one of the following conditions:
  • there are no overlined entries;

  • if there is one overlined entry on the top row, then it occurs in the anchor of a positive block;

  • if there is one overlined entry on the bottom row, then it occurs in the anchor of a negative block;

  • if there are two overlined entries, then they occur in adjacent anchors with one on the top row of the positive block and the other on the bottom row of the negative block.

Example 2.15

We consider the same Frobenius symbol as in Example 2.14. Then, all (7, 3, 3, 2)-singular overpartitions are:
$$\begin{aligned} \left( \begin{array}{c|c|c|c} {31 \; 28 \; 27 }&{} {22\; 18 \;10} &{}{ 9 \; 8 \; 7} &{} 1 \\ 29 \; 26 \; 25&{} 23 \; 22 \;20 &{} 8\; 5\; 4 &{} 3 \end{array} \right) , \left( \begin{array}{c|c|c|c} {31 \; 28 \; 27 }&{} {22\; 18\; 10 } &{}{ {\overline{9}} \; 8 \; {7}} &{} 1 \\ 29 \; 26 \; 25&{} 23 \; 22 \; 20 &{} 8\; 5\; 4 &{} 3 \end{array} \right) , \\ \left( \begin{array}{c|c|c|c} {31 \; 28 \; 27 }&{} {22\; 18\;10 } &{}{ 9 \; 8 \; 7} &{} 1 \\ 29 \; 26 \; 25&{} {\overline{23}} \; {22} \; 20 &{} 8\; 5\; 4 &{} 3 \end{array} \right) , \left( \begin{array}{c|c|c|c} {31 \; 28 \; 27 }&{} {22\; 18\;10 } &{}{ 9 \; 8 \; 7} &{} 1 \\ 29 \; 26 \; 25&{} 23 \; 22 \; 20 &{} 8\; 5\; 4 &{} {\overline{3}} \end{array} \right) , \\ \left( \begin{array}{c|c|c|c} {31 \; 28 \; 27 }&{} {22\; 18\; 10 } &{}{ {\overline{9}} \; 8 \; {7}} &{} 1 \\ 29 \; 26 \; 25&{} {\overline{23}} \; {22} \; 20 &{} 8\; 5\; 4 &{} 3 \end{array} \right) , \left( \begin{array}{c|c|c|c} {31 \; 28 \; 27 } &{} {22\; 18\; 10 } &{} {\overline{9}} \; 8 \; 7 &{} 1 \\ 29 \; 26 \; 25&{} 23 \; 22 \; 20 &{} 8\; 5\; 4 &{} {\overline{3}} \end{array} \right) . \end{aligned}$$

3 A Generalization of the Dyson Map and the Shift Map

In this section, we modify a map of Dyson, which was used to prove a symmetry of the partition function p(n) [13]. We also recall a map, the so-called shift map, from [15].

3.1 A Generalization of the Dyson Map

Let t be a positive integer and m be an integer. We define a generalized Dyson map \(d^t_{m}\) as follows:
  • Case 1: \(m\le 0\). Let \(\pi \) be a partition such that \((1, t)\in \pi \) and
    $$\begin{aligned} h_{(1,t)} \le m \quad ( \text {if and only if} \quad \pi _1-\pi '_{t} \le t-1+m). \end{aligned}$$
    (3.1)
    • Delete the first t columns in the Ferrers diagram of \(\pi \);

    • Add rows of size \(\pi '_j + m-1\) for \(1\le j\le t\) on the top of the resulting Ferrers diagram from the previous step.

    • Define \(d^t_m(\pi )\) to be the partition whose Ferrers diagram is the one obtained in the previous step.

    Here we note that \(d_m^t\) can be given explicitly as follows:
    $$\begin{aligned} d^t_m(\pi )=(\pi '_1+m-1, \ldots , \pi '_t+m-1, \pi _1-t, \pi _2-t, \ldots ), \end{aligned}$$
    which is indeed a partition since \(\pi _j\ge \pi _{j+1}\), \(\pi _j'\ge \pi '_{j+1}\) for any \(j\ge 1\), and
    $$\begin{aligned} \pi '_t+m-1\ge \pi _1-t \end{aligned}$$
    by (3.1). More details are given in Lemma 3.2 (Figs. 6, 7). For example, let \(t=2, m=-1\) and \(\pi =(4,4,2,2)\). We have
    $$\begin{aligned} h_{(1,2)}=-1 \le m=-1. \end{aligned}$$
    So,
    $$\begin{aligned} d^{2}_{-1} (\pi )= (2, 2, 2, 2). \end{aligned}$$
  • Case 2: \(m>0\). Let \(\pi \) be a partition such that \((t,1)\in \pi \) and
    $$\begin{aligned} h_{(t,1)} \ge m \quad (\text {if and only if } \quad \pi _t-\pi '_{1} \ge 1-t + m). \end{aligned}$$
    (3.2)
    Then,
    $$\begin{aligned} d^t_{m}(\pi )= \left( d^t_{-m}(\pi ')\right) '. \end{aligned}$$
    For example, let \(t=2, m=1\) and \(\pi =(4,4,2,2)\). We have
    $$\begin{aligned} h_{(2,1)}=1 \ge m=1. \end{aligned}$$
    By the definition,
    $$\begin{aligned} d^{2}_{1} (\pi )= (4 , 4). \end{aligned}$$
Fig. 6

\(d_{-1}^2(4,4,2,2)=(2,2,2,2)\)

Fig. 7

\(d_1^2(4,4,2,2)=(4,4)\)

Remark 3.1

Some facts about the generalized Dyson map are noted below.
  1. (i)

    \(|\pi | - |d^{t}_m(\pi ) |= t (|m|+1).\)

     
  2. (ii)

    For \(m\le 0\), \(d^{1}_m\) becomes the Dyson map \(d_m\) appeared in [13, 15].

     
  3. (iii)

    For \(m\le 0\), if (3.1) is not satisfied, then \(d^t_m\) is not defined. Also, for \(m>0\), if (3.2) is not satisfied, then \(d^t_m\) is not defined.

     

Lemma 3.2

For \(m\le 0\), let \(\lambda \) be a partition with \((1,t)\in \lambda \) and \(h_{(1,t)} \le m\). If \(d^{t}_m(\lambda )=\mu \ne \emptyset \), then \(\mu \) is a partition with \(\mu _j=\lambda '_j+m-1\) for \(1\le j\le t\) and \(\mu '_1=\lambda '_{t+1}+t\).

Proof

Note that \(\mu \) is a partition if and only if the parts are weakly decreasing. By the definition of \(d^t_{m}\), we know that
$$\begin{aligned} \mu _j={\left\{ \begin{array}{ll} \lambda _j'+m-1, &{} \hbox { for}\ 1\le j\le t, \\ \lambda _{j-t}-t, &{} \hbox { for}\ j>t. \end{array}\right. } \end{aligned}$$
Thus, clearly \(\mu _{j} \ge \mu _{j+1}\) for \(1\le j<t\) and \(j>t\). Also,
$$\begin{aligned} \mu _{t}-\mu _{t+1}&=(\lambda '_{t}+m-1)-(\lambda _{1}-t)\\&=\lambda '_t- \lambda _{1} +(t-1) +m \\&\ge 0. \end{aligned}$$
In addition, it is clear that \(\mu _j=\lambda '_j+m-1\) for \(1\le j\le t\) and \(\mu '_1=\lambda '_{t+1}+t\). \(\square \)
For \(m\le 0\), we first note that if \(\lambda =(t^{1-m})\), that is, \(\lambda \) is a partition with part t occurring \(1-m\) times, then \(d_{m}^t (\lambda )= \emptyset \). We now give a close look at the entries in the first column of the Frobenius symbol of \(d_{m}^t (\lambda )\) when \(\lambda \ne (t^{1-m}).\) Suppose that
$$\begin{aligned} \lambda =\left( \begin{array}{lll} a_1&{} \cdots &{} a_{\delta } \\ b_1 &{} \cdots &{} b_{\delta }\end{array} \right) . \end{aligned}$$
By Lemma 3.2,
$$\begin{aligned} \mu =\left( \begin{array}{lll} \lambda '_1+m-2 &{} \cdots &{} \\ \lambda _{t+1}'+t -1 &{} \cdots &{} \end{array} \right) =\left( \begin{array}{lll} b_1+m-1 &{} \cdots &{} \\ \gamma &{} \cdots &{} \end{array}\right) , \end{aligned}$$
(3.3)
where \(\gamma = \lambda '_{t+1} +t -1 \). If \(\delta >t\), then \(\lambda '_{t+1}=b_{t+1}+t+1\), so
$$\begin{aligned} \gamma = b_{t+1}+2t; \end{aligned}$$
if \(\delta \le t\), then \(\lambda '_{t+1}\le t\), so
$$\begin{aligned} \gamma = \lambda '_{t+1} +t -1 \le 2t-1. \end{aligned}$$

Lemma 3.3

For \(m> 0\), let \(\lambda \) be a partition with \((t,1)\in \lambda \) and \(h_{(t,1)}\ge m\). If \(d^{t}_m(\lambda )=\mu \), then \(\mu \) is a partition with \(\mu _1= \lambda _{t+1}+t\) and \(\mu '_1=\lambda _{1}-m-1\) for \(1\le j\le t\).

The proof of this lemma is similar to that of Lemma 3.2, so we omit it.

For \(m>0\), we first note that if \(\lambda =(m+1)^{t}\), then \(d_{m}^t (\lambda )= \emptyset \). Suppose that \(\lambda \ne (m+1)^{t}\) and
$$\begin{aligned} \lambda =\left( \begin{array}{lll} a_1&{} \cdots &{} a_{\delta } \\ b_1 &{} \cdots &{} b_{\delta }\end{array} \right) . \end{aligned}$$
By Lemma 3.3,
$$\begin{aligned} \mu =\left( \begin{array}{lll} \gamma &{} \cdots &{} \\ a_1- m-1 &{} \cdots &{} \end{array} \right) , \end{aligned}$$
(3.4)
where \(\gamma = \lambda _{t+1}+t-1\). If \(\delta >t\), then \(\lambda _{t+1}=a_{t+1}+t+1\), so
$$\begin{aligned} \gamma = a_{t+1} +2t; \end{aligned}$$
if \(\delta \le t\), then \(\lambda _{t+1}\le t\), so
$$\begin{aligned} \gamma = \lambda _{t+1} +t-1 \le 2t-1. \end{aligned}$$

3.2 The Shift Map

Given an integer u, a shift map \(s_u\) is defined as follows [15]:
$$\begin{aligned} \left( \begin{array}{llll} a_1 &{} a_2 &{} \cdots &{} a_{\delta } \\ b_1 &{} b_2 &{} \cdots &{} b_{\delta } \end{array} \right) {\mathop {\longrightarrow }\limits ^{s_u}} \left( \begin{array}{llll} a_1-u &{} a_2-u &{} \cdots &{} a_{\delta }-u\\ b_1+u &{} b_2+u &{} \cdots &{} b_{\delta }+u \end{array} \right) . \end{aligned}$$
(3.5)
Let \(\lambda =\left( \begin{array}{llll} a_1 &{} a_2 &{} \cdots &{} a_{\delta } \\ b_1 &{} b_2 &{} \cdots &{} b_{\delta } \end{array} \right) \) with \(a_{\delta }\ge u\), and \(\mu =s_u(\lambda )\). Then, it is clear that \(\mu \) is still a partition. Also, for \(1\le x,y \le \delta \),
$$\begin{aligned} h_{(x,y)}(\mu ) =h_{(x,y)}(\lambda )-2u. \end{aligned}$$
(3.6)
Figure 8 shows that \(s_2(5,4,2,2)=(3,2,2,2,2,2)\).
Fig. 8

Shift map

4 Lemmas

The main purpose of this section is to prove lemmas that will be used in Sect. 5.

We first introduce a necessary definition and then present lemmas.

Definition 4.1

Let \(\nu \) and \(\sigma \) be two partitions whose Frobenius symbols are
$$\begin{aligned} \nu =\left( \begin{array}{lll} a_1 &{} \cdots &{} a_{t} \\ b_1 &{} \cdots &{}b_{t} \end{array} \right) \quad \text {and}\quad \sigma =\left( \begin{array}{lll} a_{t+1} &{} \cdots &{} \\ b_{t+1} &{} \cdots &{} \end{array} \right) . \end{aligned}$$
We define the concatenation of \(\nu \) and \(\sigma \) as
$$\begin{aligned} \nu \sigma = \left( \begin{array}{llllll} a_1 &{} \cdots &{} a_{t} &{} a_{t+1} &{} \cdots &{} \\ b_1 &{} \cdots &{} b_{t} &{} b_{t+1} &{}\cdots &{} \end{array} \right) . \end{aligned}$$
Let \(\lambda =\nu \sigma \). First, note that if \(a_{t}>a_{t+1}\) and \(b_{t}>b_{t+1}\), then \(\lambda \) is indeed a partition. Even if \(\lambda \) is not a partition, we relax the definition of parts of a partition and define the xth part of \(\lambda \) as
$$\begin{aligned} \lambda _x=\nu _x+\sigma _{x-t}, \end{aligned}$$
where \(\nu _x=0\) if \(x> \ell (\nu )\), and \(\sigma _{x-t}=0\) if \(x-t>\ell (\sigma )\) or \(x-t\le 0\). In other words, \(\lambda _x\) counts the number of boxes in row x in the concatenated diagram of \(\lambda \) (see Fig. 9). We now define the hook difference at a node (xy) of \(\lambda \) as
$$\begin{aligned} h_{(x,y)}(\lambda )=\lambda _x -\lambda '_y +(x-y) \end{aligned}$$
provided \(\lambda _x\) and \(\lambda '_y\) exist. We also define the weight of \(\lambda \) by the sum of the weights of \(\nu \) and \(\sigma \), i.e.,
$$\begin{aligned} |\lambda |=|\nu |+|\sigma |. \end{aligned}$$
For instance, let
$$\begin{aligned} \nu =\left( \begin{array}{ll} 5&{} 4 \\ 4&{} 2 \end{array} \right) ,\, \sigma =\left( \begin{array}{ll} 3&{} 1 \\ 1&{} 0 \end{array} \right) . \end{aligned}$$
Then
$$\begin{aligned} \nu \sigma =\left( \begin{array}{llll} 5&{} 4 &{} 3&{} 1 \\ 4&{} 2 &{} 1&{}0 \end{array} \right) , \end{aligned}$$
which is clearly a partition. However, if we consider
$$\begin{aligned} \nu =\left( \begin{array}{ll} 5&{} 4 \\ 4&{} 2 \end{array} \right) ,\, \sigma =\left( \begin{array}{ll} 4&{} 1 \\ 3&{} 0 \end{array} \right) , \end{aligned}$$
then
$$\begin{aligned} \nu \sigma =\left( \begin{array}{llll} 5&{} 4 &{} 4 &{} 1 \\ 4&{} 2 &{} 3&{}0 \end{array} \right) , \end{aligned}$$
which is not a partition. In either case, \(|\nu \sigma |=|\nu |+|\sigma |\).
Fig. 9

Concatenation of two partitions

Although we can evaluate hook difference at any node, in this paper, we are interested in the hook differences at nodes only in the regions I–IV in Fig. 9, where
$$\begin{aligned} {\left\{ \begin{array}{ll} \;\; \;\mathrm{I}= \{ (x,y) | 1\le x,y\le t\}, \\ \;\, \mathrm{II}= \{ (x,y) | 1\le x \le t<y \le t+ a_t\}, \\ \mathrm{III}= \{(x,y) | 1\le y \le t< x \le t+b_t\}, \\ \mathrm{IV}= \{ (x,y) | t< x , y \}. \end{array}\right. } \end{aligned}$$
(4.1)
In the following lemma, we evaluate those hook differences at nodes in each of the regions I–IV in terms of \(\nu \) and \(\sigma \) for later use.

Lemma 4.2

Given partitions
$$\begin{aligned} \nu =\left( \begin{array}{lll} a_1 &{} \cdots &{} a_{t} \\ b_1 &{} \cdots &{}b_{t} \end{array} \right) \ne \emptyset \quad \text {and}\quad \sigma =\left( \begin{array}{lll} a_{t+1} &{} \cdots &{}\\ b_{t+1} &{} \cdots &{} \end{array} \right) \ne \emptyset , \end{aligned}$$
let \(\lambda =\nu \sigma \). Then
$$\begin{aligned} h_{(x,y)}(\lambda )&={\left\{ \begin{array}{ll} h_{(x,y)}(\nu ) = a_x-b_y+2 (x-y),&{} \hbox { if}\ (x,y)\in \mathrm{I}, \\ h_{(x,y)}(\nu ) -\sigma '_{y-t} = (a_x+x)-t +(x-y)-\sigma '_{y-t}, &{} \hbox { if}\ (x,y)\in \mathrm{II}, \\ h_{(x,y)}(\nu )+ \sigma _{x-t} = t -(b_y+y) +(x-y) +\sigma _{x-t}, &{} \hbox { if}\ (x,y)\in \mathrm{III}, \\ h_{(x-t,y-t)}(\sigma ) = a_{x}-b_{y}+2(x-y), &{} \text { if }(x,y)\in \mathrm{IV}. \end{array}\right. } \end{aligned}$$
(4.2)

Proof

Note that
$$\begin{aligned} \lambda _x&={\left\{ \begin{array}{ll} \nu _x =a_x+x, &{} \hbox { for}\ 1\le x \le t,\\ \nu _x+\sigma _{x-t} =t+ \sigma _{x-t}, &{} \text { for } t< x \le t+b_t, \end{array}\right. }\\ \lambda '_y&={\left\{ \begin{array}{ll} \nu '_y= b_y+y, &{} \hbox { for}\ 1\le y\le t,\\ \nu '_{y}+\sigma '_{y-t}=t +\sigma '_{y-t}, &{} \text { for }t< y \le t+a_t. \end{array}\right. } \end{aligned}$$
It is easy to prove the statement and we omit the details. \(\square \)

In the next two lemmas, we will deal with pairs of Frobenius symbols with certain conditions, and we will show that after the shift map and the Dyson map are applied to such a pair, the concatenation of the resulting pair becomes a partition.

Lemma 4.3

Given integers fgh with \(g\ge 1\), \(f\le 2g-1\), and \(h\ge f\), let \(\lambda =\nu \sigma \) with
$$\begin{aligned} \nu =\left( \begin{array}{lll} a_1 &{} \cdots &{} a_{t} \\ b_1 &{} \cdots &{}b_{t} \end{array} \right) \quad \text {and}\quad \sigma =\left( \begin{array}{lll} a_{t+1} &{} \cdots &{}\\ b_{t+1} &{} \cdots &{} \end{array} \right) \ne \emptyset , \end{aligned}$$
such that
  1. (i)

    \(h_{(t+1, t+\beta )}(\lambda ) \le f -2g+1\);

     
and if \(\nu \ne \emptyset \), then
  1. (ii)

    \(h_{(j,j+\beta -1)} (\lambda )\ge {\left\{ \begin{array}{ll} f, &{} \text { for }1\le j \le t-\beta +1, \\ f-g+1, &{} \text { for }t-\beta +2 \le j \le t , \end{array}\right. }\)

     
  2. (iii)

    \(a_{t}> a_{t+1} +g-1\), \(a_{t+1}\ge \beta -1\),

     
  3. (iv)

    \(b_{t}> b_{t+1} -g+1\), \(b_t\ge 1\).

     
If \(\mu = s_{f-g-1}(\nu ) (d^{\beta }_{f-2g+1}(\sigma ))'\), then the following are true.
  1. (a)

    \(\mu \) is a partition.

     
  2. (b)

    \(h_{(j,j+\beta -1)}(\mu )\ge -f+2g+2\) for \(1\le j\le t\), and if \((t+1, t+\beta )\in \mu \), then \(h_{(t+1,t+\beta )}(\mu )\le -f+2g+1\).

     
  3. (c)

    For an integer \(\alpha \) with \(1\le \alpha \le t\), if \(\nu \ne \emptyset \) and \(h_{(\alpha ,1)}(\nu )\ge h\), then \(h_{(\alpha , 1)} (\mu ) \ge h-2f +2g+ 2\).

     
  4. (d)

    The map from \(\lambda \) to \(\mu \) is reversible.

     
  5. (e)

    \(|\lambda |-|\mu |=(2g-f)\beta \).

     
Figure 10 sketches how to get \(\mu \) from \(\nu \) and \(\sigma \).
Fig. 10

Lemma 4.3

Proof

First, if \(\nu =\emptyset \), then \(t=0\) and \(\lambda =\sigma \), so \(h_{(t+1,t+\beta )}(\lambda )=h_{(1, \beta )} (\sigma )\). Even if \(\nu \ne \emptyset \), we see that by (4.2), \(h_{(t+1,t+\beta )}(\lambda )=h_{(1, \beta )} (\sigma )\). Thus in both cases, by Condition i) with \(f\le 2g-1\),
$$\begin{aligned} h_{(1, \beta )} (\sigma ) =h_{(t+1,t+\beta )}(\lambda ) \le f-2g+1\le 0. \end{aligned}$$
(4.3)
Hence, \(d^{\beta }_{f-2g+1}(\sigma )\) is well defined.
We now check if
$$\begin{aligned} s_{f-g-1}(\nu )=\left( \begin{array}{lll} a_1-f+g+1 &{} \cdots &{} a_{t} -f+g+1\\ b_1+f-g-1 &{} \cdots &{}b_{t} +f-g-1\end{array} \right) \end{aligned}$$
is well defined. Namely, all the entries in each row of \(s_{f-g-1}(\nu )\) are nonnegative and strictly decreasing. Since \(a_j\) and \(b_j\) are strictly decreasing, the resulting sequences \(a_j-f+g+1\) and \(b_j+f-g-1\) are strictly decreasing. For non-negativity, it suffices to check \(a_t-f+g+1\ge 0\) and \(b_t+f-g-1\ge 0\). Since
$$\begin{aligned} h_{1,\beta }(\sigma )=\sigma _1-\sigma '_{\beta }+(1-\beta )\ge \sigma _1-\sigma '_1+(1-\beta )=(a_{t+1}-b_{t+1})+(1-\beta ), \end{aligned}$$
(4.3) with \(a_{t+1}\ge \beta -1\) from Condition (iii) guarantees that
$$\begin{aligned} b_{t+1}\ge (a_{t+1}+1-\beta )+2g-f-1\ge 2g-f-1. \end{aligned}$$
So, if \(\nu \ne \emptyset \), by Condition (iv)
$$\begin{aligned} b_t +f-g -1 \ge 0. \end{aligned}$$
Also, if \(\nu \ne \emptyset \), by Condition (iii)
$$\begin{aligned} a_t\ge \beta +g-1. \end{aligned}$$
Thus
$$\begin{aligned} a_t-f+g+1&\ge \beta -f+2g \ge \beta +1\ge 2, \end{aligned}$$
where the second inequality follows from the condition that \(f\le 2g-1\). So we showed that \(s_{f-g-1}(\nu )\) is well defined.

We now prove each of the five statements.

(a) If \(\nu =\emptyset \), then it follows from Lemma 3.2 that \(\mu =(d^{\beta }_{f-2g+1}(\sigma ))'\) is a partition. Also, we note that if \(d^{\beta }_{f-2g+1}(\sigma ) = \emptyset \), then it is clear that \(\mu =s_{f-g-1}(\nu )\), which is a partition as shown above.

Now assume that \(\nu \ne \emptyset \) and \(d^{\beta }_{f-2g+1}(\sigma ) \ne \emptyset \). Let \({\tilde{\nu }}=s_{f-g-1}(\nu )\) and \({\tilde{\sigma }}=(d^{\beta }_{f-2g+1}(\sigma ))' \). Note that by (3.5),
$$\begin{aligned} {\tilde{\nu }}=\left( \begin{array}{lll} a_1-f+g+1 &{} \cdots &{} a_{t} -f+g+1 \\ b_1 + f-g-1 &{} \cdots &{} b_{t}+ f-g-1 \end{array} \right) . \end{aligned}$$
(4.4)
Also, by Lemma 3.2,
$$\begin{aligned} {\tilde{\sigma }}_1= \sigma '_{\beta +1}+\beta , \end{aligned}$$
(4.5)
and
$$\begin{aligned} {\tilde{\sigma }}'_{j} =\sigma '_{j} +(f-2g) \quad \text {for }1\le j\le \beta . \end{aligned}$$
(4.6)
So the Frobenius symbol of \({\tilde{\sigma }}\) is as follows:
$$\begin{aligned} {\tilde{\sigma }}=\left( \begin{array}{lll} \sigma '_{\beta +1}+\beta -1&{} \cdots &{} \\ \sigma '_{1} +f-2g-1 &{} \cdots &{} \end{array} \right) . \end{aligned}$$
For \(\mu \) to be a partition, it has to hold that
$$\begin{aligned} a_{t}-f+g+1>\sigma '_{\beta +1}+\beta -1 \end{aligned}$$
(4.7)
and
$$\begin{aligned} b_t+f-g-1> \sigma '_1+(f-2g)-1. \end{aligned}$$
(4.8)
We first prove (4.7). If \(\beta =1\), then \((t,t+\beta -1) \in \mathrm{I}\) in (4.1), so by (4.2) and Condition (ii),
$$\begin{aligned} h_{(t,t+\beta -1)}(\lambda )=a_t - b_t \ge f, \end{aligned}$$
so
$$\begin{aligned} (a_t-f+g+1 )-\sigma '_{2}&\ge b_t +g +1 - \sigma '_{2} \ge b_t +g+1 - b_{t+1}-1> 1, \end{aligned}$$
where the second inequality follows from \(\sigma '_2\le \sigma '_1=b_{t+1}+1\) and the last inequality follows from Condition (iv). If \(\beta >1\), then \((t,t+\beta -1)\in \mathrm{II}\) in (4.1). Thus, by (4.2) and Condition (ii),
$$\begin{aligned} h_{(t,t+\beta -1)}(\lambda )=(a_t+t)-t -(\beta -1)-\sigma '_{\beta -1} =a_t -(\beta -1)- \sigma '_{\beta -1} \ge f-g+1, \end{aligned}$$
so
$$\begin{aligned}&(a_t-f+g+1 )-(\sigma '_{\beta +1} + \beta -1)\\&\quad \ge \sigma '_{\beta -1} +(\beta -1)+2 - (\sigma '_{\beta +1} + \beta -1) \\&\quad = \sigma '_{\beta -1}-\sigma '_{\beta +1}+2> 1, \end{aligned}$$
since \(\sigma '_{\beta -1}\ge \sigma '_{\beta +1}\). This proves (4.7).
For (4.8), note that \(\sigma '_1=b_{t+1}+1\), so
$$\begin{aligned} (b_{t}+f-g-1)-(\sigma '_1+f-2g-1)&= (b_{t}-b_{t+1})+g-1>0, \end{aligned}$$
where the inequality follows from Condition (iv). Therefore, \(\mu \) is a partition.
(b) For \(1 \le j \le t-\beta +1\), i.e., \((j,j+\beta -1)\in \mathrm{I}\) in (4.1), by (4.2) and (4.4),
$$\begin{aligned} h_{(j,j+\beta -1)}(\mu )&=h_{(j,j+\beta -1)}({\tilde{\nu }})\\&= (a_{j}-f+g+1)-(b_{j+\beta -1}+f-g-1)- 2(\beta -1)\\&= (a_j - b_{j+\beta -1}) - 2(\beta -1) -2f+2g +2\\&=h_{(j, j+\beta -1)}(\nu ) -2f +2g+2\\&=h_{(j,j+\beta -1)}(\lambda ) -2f+2g+2\\&\ge -f+2g+2, \end{aligned}$$
where the last inequality follows from Condition (ii).
For \(t-\beta +2\le j\le t\), i.e., \((j,j+\beta -1)\in \mathrm{II}\) in (4.1), by (4.2), (4.4) and (4.6),
$$\begin{aligned} h_{(j,j+\beta -1)}(\mu )&= h_{(j,j+\beta -1)}({\tilde{\nu }}) - {\tilde{\sigma }}'_{j+\beta -1-t}\\&= (a_{j}-f+g+1 +j)- t - (\beta -1) - (\sigma '_{j+\beta -1-t} + f-2g)\\&= \big (a_j +j - t - (\beta -1) -\sigma '_{j+\beta -1-t} \big ) - 2f+3g +1\\&= \big (h_{(j,j+\beta -1)}(\nu ) -\sigma '_{j+\beta -1-t} \big ) - 2f+3g +1\\&=h_{(j, j+\beta -1)}(\lambda ) -2f +3g+1\\&\ge -f+2g+2, \end{aligned}$$
where the last inequality follows from Condition (ii).
Also, \((t+1, t+\beta )\in \mathrm{IV}\) in (4.1). So, by (4.2), (4.5) and (4.6),
$$\begin{aligned} h_{(t+1,t+\beta )}(\mu )&=h_{(1,\beta )}({\tilde{\sigma }}) \\&={\tilde{\sigma }}_1- {\tilde{\sigma }}'_{\beta } +(1-\beta )\\&=\sigma '_{\beta +1} +\beta - \sigma '_{\beta } - (f-2g) +(1-\beta ) \\&\le -f+2g +1, \end{aligned}$$
where the last inequality follows from \(\sigma '_{\beta +1}\le \sigma '_{\beta }\).
(c) Suppose \(h_{(\alpha ,1)}(\nu ) \ge h\) for \(1\le \alpha \le t\). Then
$$\begin{aligned} h_{(\alpha , 1)}(\mu )=h_{(\alpha , 1)}({\tilde{\nu }})= h_{(\alpha , 1)}(\nu ) -2f+2g+2 \ge h -2f+2g+2, \end{aligned}$$
where (3.6) is used for the second equality.

(d) By (b), t is uniquely determined, so we can decompose \(\mu \) into \({\tilde{\nu }}\) and \({\tilde{\sigma }}\). Since the shift map and the Dyson map are reversible, we can recover \(\nu \) and \(\sigma \) from \(\mu \).

(e) Finally, since \(|{\tilde{\nu }}|=|s_{f-g-1}(\nu )|=|\nu |\), \(|{\tilde{\sigma }}|=|d^{\beta }_{f-2g+1}(\sigma )|=|\sigma |- (2g-f)\beta \), \(|\lambda |=|\nu |+|\sigma |\), \(|\mu |=|{\tilde{\nu }}|+|{\tilde{\sigma }}|\), we have
$$\begin{aligned} |\lambda |-|\mu |=(2g-f)\beta , \end{aligned}$$
as desired. \(\square \)

Lemma 4.4

Given integers fgh with \(g -f \ge 1\), \(f \le 2g- 1\), \(h\le f\), let \(\lambda =\nu \sigma \) with
$$\begin{aligned} \nu =\left( \begin{array}{lll} a_1 &{} \cdots &{} a_{t} \\ b_1 &{} \cdots &{}b_{t} \end{array} \right) \quad \text {and}\quad \sigma =\left( \begin{array}{lll} a_{t+1} &{} \cdots &{} \\ b_{t+1} &{} \cdots &{} \end{array} \right) \ne \emptyset , \end{aligned}$$
such that
  1. (i)

    \(h_{(t+\alpha ,t+1)}(\lambda ) \ge -f+2g-1\);

     
and if \(\nu \ne \emptyset \), then
  1. (ii)

    \(h_{(j+\alpha -1,j)}(\lambda ) \le {\left\{ \begin{array}{ll} f, &{} \text {for } 1\le j \le t-\alpha +1,\\ g-1, &{} \text {for }t-\alpha +2 \le j\le t, \end{array}\right. }\)

     
  2. (iii)

    \(a_{t}>a_{t+1}+f-g+1\), \(a_t\ge 1\),

     
  3. (iv)

    \(b_t>b_{t+1}-f+g-1\), \(b_{t+1}\ge \alpha -1\).

     
If \(\mu =s_{g+1}(\nu )\, (d^{\alpha }_{2g-f-1}(\sigma ))' \), then the following are true.
  1. (a)

    \(\mu \) is a partition.

     
  2. (b)

    \(h_{(j+\alpha -1,j)}(\mu ) \le f-2g-2\) for all \(1\le j \le t\), and if \((t+\alpha , t+1)\in \mu \), then \(h_{(t+\alpha ,t+1)}(\mu ) \ge f-2g-1\).

     
  3. (c)

    For an integer \(\beta \) with \(1\le \beta \le t\), if \(\nu \ne \emptyset \) and \(h_{(1,\beta )}(\nu ) \le h\), then \( h_{(1,\beta )} (\mu ) \le h-2g-2\).

     
  4. (d)

    The map from \(\lambda \) to \(\mu \) is reversible.

     
  5. (e)

    \(|\lambda |-|\mu |=(2g-f)\alpha \).

     

Proof

Note that \(h_{(x,y)}(\lambda )=-h_{(y,x)}(\lambda ')\). We substitute \(\nu '\), \(\sigma '\), \(\alpha ,\)\(-f\), \(g-f\), and \(-h\) for \(\nu , \sigma , \beta , f, g,\) and h, respectively, in Lemma 4.3. Then we can easily check that \(\nu ', \sigma '\), \(\alpha , -f, g-f, -h\) satisfy the conditions in Lemma 4.3. Thus all the statements (a)–(e) hold true. \(\square \)

In the next two lemmas, we will discuss a lower bound of the largest part of a partition and a lower bound of the number of parts.

Lemma 4.5

Suppose that \(\lambda \) is a partition with the following non-neutral block decomposition:
$$\begin{aligned} \lambda =\left( \begin{array}{c|c|c|c|c} D_{2w}&D_{2w-1}&\cdots&D_2&D_1 \end{array}\right) . \end{aligned}$$
  1. (a)

    If \(D_1\) is negative, then \(\lambda _1\ge Kw\);

     
  2. (b)

    If \(D_1\) is positive, then \(\lambda '_1\ge Kw\).

     

Proof

(a) For \(1\le j\le 2w\), let \(x_j\) and \(y_j\) be the first and last entries in the top row of \(D_j\), and \({\tilde{x}}_j\) and \({\tilde{y}}_j\) be the first and last entries in the bottom row of \(D_j\), i.e.,
$$\begin{aligned} D_j =\begin{array}{lll} x_j &{} \cdots &{} y_j\\ {\tilde{x}}_j &{} \cdots &{} {\tilde{y}}_j \end{array}. \end{aligned}$$
First, we note that since \(D_1\) is negative, \(x_1\ge \beta -1\) by Remark 2.6.
For \(j>1\), we note that
$$\begin{aligned} y_{j}> x_{j-1}, \quad {\tilde{y}}_{j}> {\tilde{x}}_{j-1}, \end{aligned}$$
(4.9)
since \(\lambda \) is a partition. Also, we note from Lemma 2.13 that for \(j>1\), \(D_{j}\) has at least \(\beta \) columns if j is odd (i.e., \(D_j\) is negative) and at least \(\alpha \) columns if j is even (i.e., \(D_j\) is positive). Thus
$$\begin{aligned}&x_j\ge {\left\{ \begin{array}{ll} y_j+\beta -1, &{}\text { if }j\text { is odd},\\ y_j+\alpha -1, &{} \text { if }j\text { is even}, \end{array}\right. } \end{aligned}$$
(4.10)
$$\begin{aligned}&{\tilde{x}}_j\ge {\left\{ \begin{array}{ll} {\tilde{y}}_j+\beta -1, &{}\text { if }j \text {is odd},\\ {\tilde{y}}_j+\alpha -1, &{} \text { if }j\text { is even}.\end{array}\right. } \end{aligned}$$
(4.11)
For convenience, we let \(\nu =(D_{2w})\) and \(\sigma =(D_{2w-1}) \). We use induction on w.
Suppose \(w=1\). By (4.11), (4.9), we have
$$\begin{aligned} \nu '_1={\tilde{x}}_2 +1 \ge {\tilde{y}}_2+ \alpha \ge {\tilde{x}}_1+1+\alpha =\sigma '_1 +\alpha . \end{aligned}$$
(4.12)
On the other hand, since \(D_1\) is negative,
$$\begin{aligned} h_{(1,\beta )}(\sigma ) = \sigma _1-\sigma '_{\beta } +(1-\beta ) \le 1-i, \end{aligned}$$
from which we have
$$\begin{aligned} \sigma '_1\ge \sigma '_{\beta }\ge i-1+\sigma _1 +(1-\beta ) \ge i, \end{aligned}$$
(4.13)
where the last inequality follows since \(\sigma _1 \ge \beta \) by Remark 2.6. Also, since \(D_2\) is positive,
$$\begin{aligned} h_{(\alpha , 1)}(\nu ) = \nu _{\alpha }-\nu '_{1} +(\alpha -1) \ge K-i-1. \end{aligned}$$
(4.14)
Hence,
$$\begin{aligned} \lambda _1=\nu _1 \ge \nu _{\alpha }&\ge \nu '_1 +K-i-1 +(1-\alpha )\\&\ge \sigma '_1+ \alpha +K-i-1 +(1-\alpha )\\&\ge i+ \alpha +K-i-1 +(1-\alpha )=K, \end{aligned}$$
where the second, third, and last inequalities follow from (4.14), (4.12) and (4.13), respectively.
Suppose \(w>1\), and let \(\mu =(D_{2w-2}| \cdots | D_1)\). Then by the induction hypothesis, we know that
$$\begin{aligned} \mu _1\ge K(w-1). \end{aligned}$$
(4.15)
Note that \(\sigma _1=x_{2w-1}+1\) and \(\mu _1=x_{2w-2}+1\). Then, by (4.10), (4.9), we have that
$$\begin{aligned} \sigma _1 =x_{2w-1}+1 \ge y_{2w-1}+ \beta \ge x_{2w-2}+1+\beta =\mu _1+\beta \ge K(w-1)+\beta , \nonumber \\ \end{aligned}$$
(4.16)
where the last inequality follows from (4.15).
We now apply the same analysis as the \(w=1\) case to \(\nu \sigma =(D_{2w}|D_{2w-1})\). Then, the only difference happens in (4.13), which becomes
$$\begin{aligned} \sigma '_1\ge \sigma '_{\beta }\ge i-1+\sigma _1 +(1-\beta ) \ge K(w-1)+i, \end{aligned}$$
(4.17)
where the last inequality follows from (4.16). This change leads to
$$\begin{aligned} \lambda _1=\nu _1 \ge \nu _{\alpha }&\ge \nu '_1 +K-i-1 +(1-\alpha ) \\&\ge \sigma _1'+ \alpha +K-i-1 +(1-\alpha )\\&\ge K(w-1)+i+ \alpha +K-i-1 +(1-\alpha ) =Kw, \end{aligned}$$
where the second, third, and last inequalities follow from (4.14), (4.12), and (4.17), respectively.

(b) We take the conjugate of \(\lambda \), switch \(\alpha \) and \(\beta \), and then replace i by \(K-i\) in (a). Then we see that \(\lambda ' \ge Kw\). We omit the details. \(\square \)

Lemma 4.6

Suppose that \(\lambda \) is a partition with the following non-neutral block decomposition:
$$\begin{aligned} \lambda =\left( \begin{array}{c|c|c|c|c} D_{2w+1}&D_{2w}&\cdots&D_2&D_1 \end{array}\right) . \end{aligned}$$
  1. (a)

    If \(D_1\) is negative, then \(\lambda _1\ge Kw+\beta \);

     
  2. (b)

    If \(D_1\) is positive, then \(\lambda '_1\ge Kw+\alpha \).

     

Proof

(a) First, if \(w=0\), then \(\lambda =D_1\). Since \(D_1\) is negative, \(\lambda _1\ge \beta \) by Remark 2.6.

Suppose \(w>0\), and let \(\mu =(D_{2w} | \cdots | D_2 | D_1)\). By (a) in Lemma 4.5, we know that \(\mu _1\ge Kw\). Since \(\lambda = (D_{2w+1} | D_{2w} | \cdots | D_2 | D_1)\) is a Frobenius symbol, the entries in each row are strictly decreasing, so the first entry in the top row of \(D_{2w+1}\) is at least the first entry in the top row of \(D_{2w}\) plus the number of columns in \(D_{2w+1}\). Since \(D_{2w+1}\) is negative, by Lemma 2.13, there must be at least \(\beta \) columns in \(D_{2w+1}\). Thus
$$\begin{aligned} \lambda _1 \ge \mu _1+\beta \ge Kw+\beta . \end{aligned}$$
(b) We can prove this in a way similar to the proof of (a), so we omit the details. \(\square \)

5 Bijections

5.1 \((K,i,\alpha ,\beta )\)-Singular Overpartitions and Dotted Parity Blocks

For convenience, we introduce another representation of singular overpartitions, namely partitions with dotted parity blocks. Let \(\lambda \) be a \((K,i,\alpha , \beta )\)-singular overpartition. If there is exactly one overlined entry in \(\lambda \), we put a dot on the top of each of the blocks between the first non-neutral block and the block of the overlined entry. If there are two overlined entries in \(\lambda \), then we put a dot on the top of each block between the second non-neutral block and the block of the last overlined entry. In both cases, we remove the overlines from the entries. It is clear that
  1. S1.

    there are no dotted blocks, or

     
  2. S2.

    there are consecutive dotted blocks starting from the first non-neutral block, or

     
  3. S3.

    there are consecutive dotted blocks starting from the second non-neutral block.

     
For instance, if a sequence of parity blocks is EPNPN, then the following are all the dotted blocks:
  • EPNPN,

  • \(E{\dot{P}}NPN\), \(E{\dot{P}}{\dot{N}}PN\), \(E{\dot{P}}{\dot{N}}{\dot{P}}N\), \(E{\dot{P}}{\dot{N}}{\dot{P}}{\dot{N}}\),

  • \(E{P}{\dot{N}}PN\), \(E{P}{\dot{N}}{\dot{P}}N\), \(E{P}{\dot{N}}{\dot{P}}{\dot{N}}\).

Since there is a one-to-one correspondence between \((K,i, \alpha ,\beta )\)-singular overpartitions and Frobenius symbols with a sequence of parity blocks satisfying S1, or S2, or S3, we will use the latter form in this section.

For a positive integer m, let \({\dot{p}}^{-}_{K,i,\alpha , \beta }(m,n)\) (or \({\dot{p}}^{+}_{K,i,\alpha , \beta }(m,n)\)) be the number of partitions of n with exactly m dotted parity blocks and the last block negative (or positive, resp.).

Theorem 5.1

For \(m\ge 1\) and \(n \ge 0\),
$$\begin{aligned}&{\dot{p}}^{-}_{K,i,\alpha , \beta }(m,n) \nonumber \\&\quad = p\left( n -\left( K\left\lfloor \frac{m}{2}\right\rfloor ^2 + i\left\lfloor \frac{m}{2}\right\rfloor \right) \alpha - \left( K\left\lceil \frac{m}{2}\right\rceil ^2 - (K-i)\left\lceil \frac{m}{2}\right\rceil \right) \beta \right) , \end{aligned}$$
(5.1)
$$\begin{aligned}&{\dot{p}}^{+}_{K,i,\alpha , \beta }(m,n) \nonumber \\&\quad = p\left( n -\left( K\left\lceil \frac{m}{2}\right\rceil ^2 - i\left\lceil \frac{m}{2}\right\rceil \right) \alpha - \left( K\left\lfloor \frac{m}{2}\right\rfloor ^2 + (K-i)\left\lfloor \frac{m}{2}\right\rfloor \right) \beta \right) . \end{aligned}$$
(5.2)

The proof of Theorem 5.1 will be given in Sect. 5.2. When \(\alpha =\beta \), Theorem 5.1 yields the following theorem.

Theorem 5.2

For \(m \ge 1\) and \(n \ge 0\),
$$\begin{aligned} {\dot{p}}^{-}_{K,i,\alpha , \alpha }(m,n)&=p\left( n-\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) -\alpha i m \right) , \\ {\dot{p}}^{+}_{K,i,\alpha , \alpha }(m,n)&=p\left( n-\alpha K\left( {\begin{array}{c}m+1\\ 2\end{array}}\right) +\alpha i m \right) . \end{aligned}$$

We note that when \(\alpha =1\), Theorem 5.2 yields Theorem 3.1 in [15].

5.2 The Bijection \(\psi ^{\alpha ,\beta }_m\)

In this section, we will prove Theorem 5.1 by constructing a bijection between partitions with dotted parity blocks and ordinary partitions. We will prove only (5.1). The proof of (5.2) will be similar, so it will be omitted.

Let us denote the set of partitions of n by \({\mathcal {P}}(n)\). Also, let \(\mathcal {\dot{P}}^{-}_{K,i,\alpha ,\beta }(m,n)\) be the set of partitions of n with exactly m dotted parity blocks with the last block negative. We will construct a bijection \(\psi ^{\alpha ,\beta }_m\) from \(\mathcal {\dot{P}}^{-}_{K,i,\alpha , \beta }(m,n)\) to \({\mathcal {P}}(N)\), where
$$\begin{aligned} N={\left\{ \begin{array}{ll} n- (Ku^2+iu)\alpha - \left( Ku^2-(K-i)u\right) \beta , &{} \hbox { if}\ m=2u,\\ n- (Ku^2+iu)\alpha - \left( K(u+1)^2-(K-i)(u+1)\right) \beta , &{} \hbox { if}\ m=2u+1.\end{array}\right. } \end{aligned}$$
(5.3)
Let \(\lambda \) be a partition in \(\mathcal {\dot{P}}^{-}_{K,i,\alpha ,\beta }(m,n)\). First let \(D_1\) be the union of the last dotted block and the blocks on the right of the last dotted block if any. From right to left, denote each of the unchosen dotted blocks by \(D_v\) for \(1<v \le m\). Let \(D_{m+1}\) be the union of the blocks on the left of \(D_{m}\) if any.
Let us recall the (5, 2, 2, 2)-singular overpartition from Example 2.11:
$$\begin{aligned} \lambda =\left( \begin{array}{c|cc|cc|ccc|cc} 31 &{} 28 &{} 27 &{} 22 &{} 18 &{} 9 &{} 8 &{} 7 &{} 1 &{} 0 \\ 29 &{} 26 &{} 25 &{} 23 &{} 22 &{} 8 &{} 5 &{} 4 &{} 1 &{} 0 \end{array} \right) , \end{aligned}$$
with its sequence of dotted blocks \(EP{\dot{N}}{\dot{P}}{\dot{N}}\). Then we have
$$\begin{aligned} D_4=\left( \begin{array}{c|cc} 31 &{} 28 &{} 27\\ 29 &{} 26 &{} 25 \end{array}\right) ,~ D_3=\left( \begin{array}{ll} 22 &{} 18\\ 23 &{} 22 \end{array}\right) ,~ D_2=\left( \begin{array}{lll} 9 &{} 8 &{} 7\\ 8 &{} 5 &{} 4 \end{array}\right) ,~ D_1=\left( \begin{array}{ll} 1&{} 0 \\ 1&{} 0 \end{array}\right) . \end{aligned}$$
We then define \(\Gamma _{1},\ldots ,\Gamma _{m+1}\) and \(\psi ^{\alpha ,\beta }_{m}(\lambda )\) as follows:
  • Set \(\Gamma _{1}=D_1\).

  • For \(1\le v \le m\), set
    $$\begin{aligned} \Gamma _{v+1}={\left\{ \begin{array}{ll} s_{-i-wK }(D_{v+1})\, (d^{\beta }_{1-i-(v-1)K}(\Gamma _{v}) )', &{} \text{ if } v=2w+1 \text{ for } \text{ some } w\ge 0,\\ s_{ wK }(D_{v+1})\, (d^{\alpha }_{-1+i+(v-1)K}(\Gamma _{v}) )', &{} \text{ if } v=2w \text{ for } \text{ some } \,w>0.\\ \end{array}\right. } \end{aligned}$$
  • Define \(\psi ^{\alpha ,\beta }_{m}(\lambda )=\Gamma _{m+1}\).

Now we will inductively show that for each \(1 \le v \le m\), \(\Gamma _{v}\) is a partition satisfying
$$\begin{aligned} h_{(1,\beta )}(\Gamma _{v})&\le 1-i-(v-1)K, \quad \hbox { if}\ v=2w+1, \end{aligned}$$
(5.4)
$$\begin{aligned} h_{(\alpha ,1)}(\Gamma _{v})&\ge -1+ i+(v-1)K, \; \text { if }v=2w. \end{aligned}$$
(5.5)
First, since \(\Gamma _1=D_1\) and its first column is \((K,i,\alpha ,\beta )\)-negative, \(\Gamma _1\) is a partition satisfying (5.4).
Assume that for \(1\le v<m\), \(\Gamma _{v}\) is well defined and satisfies (5.4) or (5.5). We now prove that \(\Gamma _{v+1}\) is a partition satisfying (5.4) or (5.5). Let t be the number of columns in \(D_{v+1}\).
  • Case 1: Suppose \(v=2w+1\) for some \(w\ge 0\). Then we can write \(\Gamma _{v+1}\) as
    $$\begin{aligned} \Gamma _{v+1}=s_{-i-wK}(D_{v+1})\, (d^{\beta }_{1-i-(v-1)K}(\Gamma _{v}))'. \end{aligned}$$
    In Lemma 4.3, set \(f=2-i\), \(g=wK+1\), \(h=K-i-1\), and \(\nu =D_{v+1}\), \(\sigma =\Gamma _{v}\). Clearly fgh satisfy \(g\ge 1\), \(f\le 2g -1\), \(h\ge f\), since \(1\le i<K/2\). Let us check the four conditions of Lemma 4.3. First, \((t+1,t+\beta )\in \mathrm{IV}\) in (4.1). So, by (4.2),
    $$\begin{aligned} h_{(t+1,t+\beta )}(D_{v+1}\Gamma _v) =h_{(1,\beta )}(\Gamma _{v}). \end{aligned}$$
    Thus, by the induction hypothesis (5.4),
    $$\begin{aligned} h_{(t+1,t+\beta )}(D_{v+1}\Gamma _v) =h_{(1,\beta )}(\Gamma _{v})\le 1-i-2wK=f-2g+1, \end{aligned}$$
    so Condition (i) holds true. Let us verify Condition (ii) of the lemma. To that end, we have to consider two different regions where a node \((j,j+\beta -1)\) is placed, namely \(j\le t-\beta +1\) and \(j\ge t-\beta +2\), i.e., \((j,j+\beta -1)\in \mathrm{I}\) and \((j,j+\beta -1)\in \mathrm{II}\) in (4.1). In Fig. 11, the node a falls in the first case and the node b falls in the second case. The nodes c and d will be discussed later in Case 2. First note that the hook difference at the node a is unchanged after \(D_v\) and \(\Gamma _{v-1}\) are merged to become \(\Gamma _{v}\), namely,
    $$\begin{aligned} h_{(j,j+\beta -1)}(D_{v+1}\Gamma _v) = h_{(j,j+\beta -1)}(D_{v+1}D_v\Gamma _{v-1}). \end{aligned}$$
    (5.6)
    On the other hand, since
    $$\begin{aligned} \Gamma _v=s_{wK}(D_{v})\, \big (d^{\alpha }_{-1+i+(v-2)K}(\Gamma _{v-1})\big )', \end{aligned}$$
    and \(D_v\) is negative, i.e., there are at least \(\beta \) columns, the hook difference at the node b is affected only by the shift map \(s_{wK}\) that is applied to \(D_v\), namely,
    $$\begin{aligned} h_{(j,j+\beta -1)}(D_{v+1}\Gamma _v)&=h_{(j,j+\beta -1)}(D_{v+1}D_{v}\Gamma _{v-1})-wK \nonumber \\&=h_{(j,j+\beta -1)}(D_{v+1}D_{v}\Gamma _{v-1})- (g-1). \end{aligned}$$
    (5.7)
    Also, we note that \(D_{v+1}\) cannot be \((K,i, \alpha , \beta )\)-negative because the last dotted block is negative and the signs of blocks are alternating. Thus,
    $$\begin{aligned} h_{(j,j+\beta -1)}(D_{v+1}D_v\Gamma _{v-1}) \ge 2-i =f. \end{aligned}$$
    (5.8)
    Therefore, by (5.6), (5.7), and (5.8),
    $$\begin{aligned} h_{(j,j+\beta -1)}(D_{v+1}\Gamma _v) \ge {\left\{ \begin{array}{ll} f, &{} \hbox { for}\ 1\le j\le t-\beta +1,\\ f- g+1, &{} \hbox { for}\ t-\beta +2\le j\le t, \end{array}\right. } \end{aligned}$$
    which verifies that Condition (ii) holds true. Lastly, let \(\begin{array}{l} x_1 \\ x_2 \end{array}\) and \(\begin{array}{l} z_1\\ z_2\end{array}\) be the last column of \(D_{v+1}\) and the first column of \(D_{v}\), respectively, i.e.,
    $$\begin{aligned} (D_{v+1} | D_{v})=\left( \begin{array}{cc|cc} \cdots &{} x_1 &{} z_1 &{} \cdots \\ \cdots &{} x_2 &{} z_2 &{} \cdots \end{array}\right) . \end{aligned}$$
    Since \(D_{v+1}D_{v}\) forms a Frobenius symbol, we have
    $$\begin{aligned} x_1>z_1, \quad x_2>z_2. \end{aligned}$$
    By Lemma 4.6 (a), we know that \(z_1 \ge wK+\beta -1\). Since
    $$\begin{aligned} \Gamma _{v}= {\left\{ \begin{array}{ll} D_1, &{} \hbox { for}\ v=1, \\ s_{wK}(D_{v}) (d^{\alpha }_{-1+i +(v-2)K}(\Gamma _{v-1}))', &{} \hbox { for}\ v>2, \end{array}\right. } \end{aligned}$$
    the first column of \(\Gamma _{v}\) is \(\begin{array}{l} z_1-wK\\ z_2+wK \end{array}.\) Thus, we have
    $$\begin{aligned} x_1>(z_1-wK)+ (wK+1)-1&\text { and } z_1-wK\ge \beta -1,\\ x_2>(z_2+wK)- (wK+1)+1&\text { and } x_2 >z_2 \ge 0, \end{aligned}$$
    which verify Conditions (iii) and (iv). Since all the four conditions in Lemma 4.3 are satisfied, by Statement (a) of Lemma 4.3, \(\Gamma _{v+1}\) is a partition. Also, since \(v<m\), \(D_{v+1}\ne \emptyset \) is indeed a positive block, so
    $$\begin{aligned} h_{(\alpha ,1)}(D_{v+1}) \ge K-i-1=h. \end{aligned}$$
    Thus, by Statement (c) of Lemma 4.3,
    $$\begin{aligned} h_{(\alpha ,1)}(\Gamma _{v+1})\ge h-2f+2g+2=-1+i+(2w+1)K, \end{aligned}$$
    which verifies (5.5).
  • Case 2: Suppose \(v=2w\) for some \(w\ge 1\). Then we can write \(\Gamma _{v+1}\) as
    $$\begin{aligned} \Gamma _{v+1}=s_{wK}(D_{v+1})\,(d^{\alpha }_{-1+i+(v-1)K}(\Gamma _{v}))'. \end{aligned}$$
    In Lemma 4.4, set \(f=K-i-2\), \(g=wK-1\), \(h=1-i\), and \(\nu =D_{v+1}\), \(\sigma =\Gamma _{v}\). Clearly, fg and h satisfy \(g-f \ge 1\), \(f\le 2g-1\), \(h\le f\). Next, let us verify the four conditions of Lemma 4.4. First, note that \((t+\alpha , t+1)\in \mathrm{IV}\) in (4.1). So, by (4.2), we know that
    $$\begin{aligned} h_{(t+\alpha ,t+1)}(D_{v+1}\Gamma _v) =h_{(\alpha ,1)}(\Gamma _{v}). \end{aligned}$$
    Thus, by the induction hypothesis (5.5),
    $$\begin{aligned} h_{(t+\alpha ,t+1)}(D_{v+1}\Gamma _v)&=h_{(\alpha ,1)}(\Gamma _{v})\ge -1+i+(2w-1)K\\&=-f+2g-1, \end{aligned}$$
    so Condition (i) holds true. For Condition (ii), we have to consider two different regions where a node \((j+\alpha -1,j)\) is placed, namely \(j\le t-\alpha +1\) and \(j\ge t-\alpha +2\), i.e., \((j+\alpha -1,j)\in \mathrm{I}\) and \((j+\alpha -1,j)\in \mathrm{III}\) in (4.1). In Fig. 11, the node c falls in the first case and the node d falls in the second case. First note that the hook difference at the node c is unchanged after \(D_v\) and \(\Gamma _{v-1}\) are merged to become \(\Gamma _{v}\), namely,
    $$\begin{aligned} h_{(j+\alpha -1,j)}(D_{v+1}\Gamma _v) = h_{(j+\alpha -1,j)}(D_{v+1}D_v\Gamma _{v-1}). \end{aligned}$$
    (5.9)
    On the other hand, since
    $$\begin{aligned} \Gamma _v=s_{-i-(w-1)K}(D_{v})\, (d^{\beta }_{1-i-(v-2)K}(\Gamma _{v-1}))', \end{aligned}$$
    and \(D_v\) is positive, i.e., there are at least \(\alpha \) columns, the hook difference at the node d is affected only by the shift map \(s_{-i-(w-1)K}\) that is applied to \(D_v\), namely,
    $$\begin{aligned} h_{(j+\alpha -1,j)}(D_{v+1}\Gamma _v)&=h_{(j+\alpha -1,j)}(D_{v+1}D_{v}\Gamma _{v-1})+i+(w-1)K \nonumber \\&=h_{(j+\alpha -1,j)}(D_{v+1}D_{v}\Gamma _{v-1})-f+g-1. \end{aligned}$$
    (5.10)
    Also, we note that \(D_{v+1}\) cannot be \((K,i, \alpha , \beta )\)-positive because the last dotted block is negative and the signs of blocks are alternating. Thus,
    $$\begin{aligned} h_{(j+\alpha -1,j)}(D_{v+1}D_v\Gamma _{v-1}) \le K-i-2 =f. \end{aligned}$$
    (5.11)
    Therefore, by (5.9), (5.10), and (5.11),
    $$\begin{aligned} h_{(j+\alpha -1,j) }(D_{v+1}\Gamma _v) \le {\left\{ \begin{array}{ll} f, &{} \hbox { for}\ 1\le j\le t-\alpha +1,\\ g-1, &{} \hbox { for}\ t-\alpha +2\le j\le t, \end{array}\right. } \end{aligned}$$
    which verifies that Condition (ii) holds true. Lastly, \(D_{v+1}D_{v}\) forms a Frobenius symbol. Thus, in the same way as in Case 1, Conditions (iii) and (iv) in Lemma 4.4 can be verified. We omit the details. Therefore, by Statements (a) and (c) of Lemma 4.4, \(\Gamma _{v+1}\) is a partition satisfying (5.4).
Fig. 11

\(D_{v+1}D_v\Gamma _{v-1} \rightarrow D_{v+1} \Gamma _v\)

We now have that \(\Gamma _m\) is a partition satisfying (5.4) or (5.5) from the induction. Also, \(D_{m+1}\) is a partition. We can easily check that \(D_{m+1}\) and \(\Gamma _m\) satisfy the conditions for Lemmas 4.3 or 4.4. Therefore, \(\Gamma _{m+1}\) is a partition by Statement (a) of each lemma. Here we note that all the arguments for \(v<m\) hold for \(v=m\) except that if the first column of \(D_{m+1}\) is neutral, then Statement (c) does not hold. However, Statement (c) is not needed to complete our proof, for what we need to prove is that \(\Gamma _{m+1}\) is a partition.

Let us then check the weight difference. By Statement (e) of each of Lemmas 4.3 and 4.4, we have
$$\begin{aligned} |D_{v+1}\Gamma _{v}|-|\Gamma _{v+1}|={\left\{ \begin{array}{ll} \left( i+(v-1)K\right) \beta , &{} \hbox { if}\ v=2w+1, \\ \left( i+(v-1)K\right) \alpha , &{} \hbox { if}\ v=2w, \end{array}\right. } \end{aligned}$$
(5.12)
for \(v=1,\ldots , m\). By (5.12), we have
$$\begin{aligned} |\lambda |&=|D_{m+1} D_m \cdots D_5D_{4} D_{3} D_{2} D_{1} | \\&=|D_{m+1} D_m \cdots D_5D_{4} D_{3} D_{2} \Gamma _1| \\&=|D_{m+1} D_m \cdots D_5D_{4} D_{3} \Gamma _2|+i \beta \\&=|D_{m+1} D_m \cdots D_5D_{4} \Gamma _3|+(i+K)\alpha +i\beta \\&=|D_{m+1} D_m \cdots D_5 \Gamma _4 |+ (i+2K)\beta + (i+K)\alpha +i\beta \\&\;\; \vdots \\&=|{\Gamma }_{m+1}| \\&\quad + {\left\{ \begin{array}{ll} \sum \limits _{v=1}^{u} \bigg (\big ( i + (2v-1)K \big )\alpha + \big (i+(2v-2)K\big )\beta \bigg ), &{} \hbox { if}\ m=2u,\\ (i+2uK)\beta +\sum \limits _{v=1}^{u} \bigg (\big ( i + (2v-1)K \big )\alpha + \big (i+(2v-2)K \big )\beta \bigg ), &{} \text { if }m=2u+1. \end{array}\right. } \end{aligned}$$
Thus
$$\begin{aligned} |{\Gamma }_{m+1}| = {\left\{ \begin{array}{ll} |\lambda |- (Ku^2+iu)\alpha - (Ku^2-Ku+iu)\beta , &{} \hbox { if}\ m=2u,\\ |\lambda |- (Ku^2+iu)\alpha - (Ku^2+Ku+iu+i)\beta , &{} \text { if }m=2u+1. \end{array}\right. } \end{aligned}$$
This shows that \(\psi ^{\alpha ,\beta }_m\) is a map from \(\mathcal {\dot{P}}^{-}_{K,i,\alpha ,\beta }(m,n)\) to \({\mathcal {P}}(N)\), where N is given in (5.3) as desired.

In addition, by Statement (d) of Lemma 4.3 and Lemma 4.4, each process of producing \(\Gamma _{v+1}\) is reversible. Therefore, \(\psi ^{\alpha ,\beta }_m\) is indeed a bijection.

Example 5.3

Consider a (5, 2, 2, 2)-singular overpartition
$$\begin{aligned} \lambda =\left( \begin{array}{c|cc|cc|ccc|cc} 31 &{} 28 &{} 27 &{} 22 &{} 18 &{} 9 &{} 8 &{} 7 &{} 1 &{} 0 \\ 29 &{} 26 &{} 25 &{} 23 &{} 22 &{} 8 &{} 5 &{} 4 &{} 1 &{} 0 \end{array} \right) , \end{aligned}$$
with its sequence of dotted blocks \(EP{\dot{N}}{\dot{P}}{\dot{N}}\). Note that \(K=5\), \(i=\alpha =\beta =2\), and \(m=3\). We have the following \(\Gamma _v\) for \(v=1,2,3,4\):
where the dashed line is put to indicate the concatenated two arrays in each \(\Gamma _v\). Here \({\Gamma }_4\) is the ordinary partition corresponding to the (5, 2, 2, 2)-singular overpartition \(\lambda \). Lastly, we check their weight difference
$$\begin{aligned} |\lambda |-|\Gamma _4|=304-262=42, \end{aligned}$$
which matches \((Ku^2+iu)\alpha + (Ku^2+Ku+iu+i)\beta \) as desired.

6 Results

In this section, we will relate \((K,i,\alpha ,\beta )\)-singular overpartitions with ordinary partitions.

For a \((K,i,\alpha , \beta )\)-singular overpartition with overlined entries in anchors, if the first overlined entry occurs in the mth block, then the mth block can be negative or positive, and the next anchor can have an overlined entry if exists. In all these four cases, i.e., only one overlined entry in either a negative or a positive block, or two overlined entries in two consecutive and opposite parity blocks, we see from the definition of dotted parity blocks given in the beginning of Sect. 5.1 that such singular overpartitions are partitions with exactly m dotted parity blocks. Thus,
$$\begin{aligned} {\overline{Q}}_{K,i,\alpha ,\beta }(m,n)={\dot{p}}^-_{K,i,\alpha ,\beta }(m,n)+{\dot{p}}^+_{K,i,\alpha ,\beta }(m,n), \end{aligned}$$
(6.1)
where \({\overline{Q}}_{K,i,\alpha ,\beta }(m,n)\) is the number of \((K,i,\alpha , \beta )\)-singular overpartitions of n with an overlined entry in its mth anchor, which is defined before Theorem 1.2 in Introduction.

Theorem 6.1

For \(m\ge 1\) and \(n \ge 0\),
$$\begin{aligned}&{\overline{Q}}_{K,i, \alpha , \beta } (m, n)\\&\quad =p\left( n-\left( K \left\lceil \frac{m}{2}\right\rceil ^2 - i\left\lceil \frac{m}{2}\right\rceil \right) \alpha - \left( K \left\lfloor \frac{m}{2}\right\rfloor ^2 + (K-i)\left\lfloor \frac{m}{2}\right\rfloor \right) \beta \right) \\&\qquad +p\left( n-\left( K \left\lfloor \frac{m}{2}\right\rfloor ^2 + i\left\lfloor \frac{m}{2}\right\rfloor \right) \alpha - \left( K \left\lceil \frac{m}{2}\right\rceil ^2 - (K-i)\left\lceil \frac{m}{2}\right\rceil \right) \beta \right) , \end{aligned}$$
where p(N) denotes the number of ordinary partitions of N with \(p(0)=1\) and \(p(N)=0\) for \(N<0\).

Proof

This theorem follows from (6.1) and Theorem 5.1. \(\square \)

Remark 6.2

Some facts about \({\overline{Q}}_{K,i, \alpha , \beta } (m, n)\) are noted below.
  1. (i)
    Since \(\left\lfloor \frac{m}{2}\right\rfloor = -\left\lceil \frac{-m}{2}\right\rceil \) and \(\left\lceil \frac{m}{2}\right\rceil = -\left\lfloor \frac{-m}{2}\right\rfloor \), we have
    $$\begin{aligned}&{\overline{Q}}_{K,i, \alpha , \beta } (m, n)\\&~=p\left( n-\left( K \left\lceil \frac{m}{2}\right\rceil ^2 - i\left\lceil \frac{m}{2}\right\rceil \right) \alpha - \left( K \left\lfloor \frac{m}{2}\right\rfloor ^2 + (K-i)\left\lfloor \frac{m}{2}\right\rfloor \right) \beta \right) \\&\quad +p\left( n-\left( K \left\lceil \frac{-m}{2}\right\rceil ^2 - i\left\lceil \frac{-m}{2}\right\rceil \right) \alpha - \left( K \left\lfloor \frac{-m}{2}\right\rfloor ^2 + (K-i)\left\lfloor \frac{-m}{2}\right\rfloor \right) \beta \right) \end{aligned}$$
    for \(m\ge 1\) and \(n\ge 0\).
     
  2. (ii)
    Each ordinary partition of n can be regarded as a \((K,i,\alpha ,\beta )\)-singular overpartition without any overlined entries. Thus
    $$\begin{aligned} {\overline{Q}}_{K,i,\alpha ,\beta }(0,n)=p(n). \end{aligned}$$
     
Let us recall \({\overline{Q}}_{K,i,\alpha ,\beta }(n)\):
$$\begin{aligned} {\overline{Q}}_{K,i,\alpha ,\beta }(n)=\sum _{m=0}^{\infty } {\overline{Q}}_{K,i,\alpha ,\beta }(m,n). \end{aligned}$$
(6.2)

Theorem 6.3

We have
$$\begin{aligned}&\sum _{n=0}^{\infty } {\overline{Q}}_{K,i,\alpha ,\beta }(n) q^n \\&\quad =\frac{(q^{2K(\alpha +\beta )}, -q^{(K+i)\alpha +i\beta }, -q^{(K-i)\alpha +(2K-i)\beta } ;q^{2K(\alpha +\beta )})_{\infty }}{(q;q)_{\infty }}\\&\qquad +q^{i\beta } \frac{(q^{2K(\alpha +\beta )}, -q^{(K-i)\alpha -i\beta }, -q^{(K+i)\alpha +(2K+i)\beta } ;q^{2K(\alpha +\beta )})_{\infty }}{(q;q)_{\infty }}, \end{aligned}$$
where \((a_1, a_2, \dots , a_M ;q)_{\infty }=(a_1;q)_{\infty }(a_2;q)_{\infty }\cdots (a_M;q)_{\infty }\).

Proof

By (6.2), Theorem 6.1, and Remark 6.2,
$$\begin{aligned}&{\overline{Q}}_{K,i,\alpha ,\beta }(n) \\&\quad =\sum _{m=-\infty }^{\infty } p\left( n -\left( K\left\lfloor \frac{m}{2}\right\rfloor ^2 + i\left\lfloor \frac{m}{2}\right\rfloor \right) \alpha - \left( K\left\lceil \frac{m}{2}\right\rceil ^2 - (K-i)\left\lceil \frac{m}{2}\right\rceil \right) \beta \right) \\&\quad =\sum _{u=-\infty }^{\infty } p\left( n- (Ku^2+iu)\alpha - (Ku^2-Ku+iu)\beta \right) \qquad \,\,\,\quad (m=2u)\\&\qquad +\sum _{u=-\infty }^{\infty } p\left( n- (Ku^2+iu)\alpha - (Ku^2+Ku+iu+i)\beta \right) \quad (m=2u+1). \end{aligned}$$
Thus,
$$\begin{aligned}&\sum _{n=0}^{\infty } {\overline{Q}}_{K,i,\alpha ,\beta }(n) q^n \\&\quad =\sum _{n=0}^{\infty } \sum _{u=-\infty }^{\infty } p\left( n- (Ku^2+iu)\alpha - (Ku^2-Ku+iu)\beta \right) q^n \\&\qquad +\sum _{n=0}^{\infty }\sum _{u=-\infty }^{\infty } p\left( n- (Ku^2+iu)\alpha - (Ku^2+Ku+iu+i)\beta \right) q^n\\&\quad =\frac{1}{(q;q)_{\infty }} \sum _{u=-\infty }^{\infty } q^{(Ku^2+iu)\alpha +(Ku^2-Ku+iu)\beta } \\&\qquad + \frac{1}{(q;q)_{\infty }} \sum _{u=-\infty }^{\infty } q^{(Ku^2+iu)\alpha +(Ku^2+Ku+iu+i)\beta } \\&\quad =\frac{(q^{2K(\alpha +\beta )}, -q^{(K+i)\alpha +i\beta }, -q^{(K-i)\alpha +(2K-i)\beta } ;q^{2K(\alpha +\beta )})_{\infty }}{(q;q)_{\infty }}\\&\qquad +q^{i\beta } \frac{(q^{2K(\alpha +\beta )}, -q^{(K+i)\alpha +(2K+i)\beta }, -q^{(K-i)\alpha -i\beta } ;q^{2K(\alpha +\beta )})_{\infty }}{(q;q)_{\infty }}, \end{aligned}$$
where the last equality follows from Jacobi’s triple product identity [3, 17]. \(\square \)

6.1 Proof of Theorem 1.1

When \(\alpha =\beta \), Theorem 6.3 can be simplified further. By Theorem 5.2, we have
$$\begin{aligned} {\overline{Q}}_{K,i,\alpha ,\alpha }(n)&=\sum _{m=-\infty }^{\infty } p\left( n-\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) -\alpha i m\right) , \end{aligned}$$
where we use the fact \(\left( {\begin{array}{c}m+1\\ 2\end{array}}\right) =\left( {\begin{array}{c}-m\\ 2\end{array}}\right) \). Thus,
$$\begin{aligned}&\sum _{n=0}^{\infty } {\overline{Q}}_{K,i,\alpha ,\alpha }(n) q^n \nonumber \\&\quad = \sum _{n=0}^{\infty } \sum _{m=-\infty }^{\infty } p\left( n-\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) -\alpha i m\right) q^n \nonumber \\&\quad =\sum _{m=-\infty }^{\infty } q^{\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) +\alpha i m} \sum _{n=0}^{\infty } p\left( n-\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) -\alpha i m\right) q^{n-\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) -\alpha i m} \nonumber \\&\quad = \sum _{m=-\infty }^{\infty } q^{\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) +\alpha i m} \sum _{n=0}^{\infty } p(n ) q^{n} \nonumber \\&\quad =\frac{1}{(q;q)_{\infty }} \sum _{m=-\infty }^{\infty } q^{\alpha K\left( {\begin{array}{c}m\\ 2\end{array}}\right) +\alpha i m} \nonumber \\&\quad =\frac{(-q^{i\alpha }, -q^{(K-i)\alpha }, q^{K\alpha };q^{K\alpha })_{\infty }}{(q;q)_{\infty }}, \end{aligned}$$
(6.3)
where the last equality follows from Jacobi’s triple product identity [3, 17]. This proves Theorem 1.1.

We easily see that the right hand side of (6.3) is the generating function of overpartitions in which parts \(\not \equiv 0\) mod \(K\alpha \) and only parts \(\equiv \pm i\alpha \) mod \(K\alpha \) may be overlined.

6.2 Proof of Theorem 1.2

Theorem 1.2 follows immediately from (6.1) and Theorem 5.2.

7 Remarks

We provide a few remarks. First, for a positive integer \(k>1\), the case when \(i= k\) and \(K= 2k\) is investigated by Bressoud in [8]. In [16], when \(K=3, i=\alpha =\beta =1\), further refined cases were studied.

Finally, let \({p}^{E}_{K,i,\alpha ,\beta }(n)\) be the number of partitions of n without any signed blocks. Since \({\overline{Q}}_{K,i,\alpha ,\beta }(m,n)\) counts the number of singular overpartitions of n with an overlined entry in the mth anchor, it is the same as the number of partitions of n with at least m signed blocks. By the sieving method, we have
$$\begin{aligned}&p^E_{K,i,\alpha ,\beta }(n)=p(n)+ \sum _{m\ge 1}(-1)^m {\overline{Q}}_{K,i,\alpha ,\beta }(m,n), \end{aligned}$$
from which we can deduce Theorem 2 in [6].

Notes

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Authors and Affiliations

  1. 1.Department of Mathematics EducationKangwon National UniversityChuncheonRepublic of Korea
  2. 2.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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