# Singular Overpartitions and Partitions with Prescribed Hook Differences

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## 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, (*K*, *i*)-singular overpartitions for integers *K*, *i* 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 *K*, *i* with \(1\le i<K/2\), Andrews defined a subclass of singular overpartitions with some restrictions subject to *K* and *i*, namely (*K*, *i*)-singular overpartitions. He then showed interesting combinatorial and arithmetic properties of (*K*, *i*)-singular overpartitions. As seen in [5], (*K*, *i*)-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 (*K*, *i*)-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 *K*, *i*, \(\alpha \) 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 note that \({\overline{Q}}_{K,i, 1, 1}(n)\) becomes the number of (*K*, *i*)-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 *m*th anchor. Again, because of the complexity, anchors are defined in Sect. 2. Then we have the following theorem.

### Theorem 1.2

*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

*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

*j*th row has as many boxes as the

*j*th part \(\lambda _j\). The box in row

*x*and column

*y*of the Ferrers diagram is called node (

*x*,

*y*). If a node (

*x*,

*y*) 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 (

*x*,

*y*) with \(x=y+k\) [6]. Figure 1 shows some diagonals on the partition (5, 4, 2, 2).

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

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

*x*with \(0\le x \le \beta -1\),

### Proof

- Case 1: \((j,j+\beta -1)\in \lambda \). If \((j+x+\alpha -1, j+x) \in \lambda \), thenwhere 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$$\begin{aligned} h_{(j+x+\alpha -1, j+x)}&=\lambda _{j+x+\alpha -1}-\lambda '_{j+x} +\alpha -1 \\&\le \lambda _{j}-\lambda '_{j+x} +\alpha -1\\&\le \lambda _j-\lambda '_{j+\beta -1}+\alpha -1 \\&=h_{(j,j+\beta -1)} + (\alpha +\beta -2)\\&\le 1-i + (\alpha + \beta )-2\\&\le K-i-2, \end{aligned}$$where the left inequality follows from (2.2) and the right inequality follows from \(1\le i <K/2\).$$\begin{aligned} h_{(j+x+\alpha -1, j+x)} \le 0 \le K-i-2, \end{aligned}$$
- Case 2: \((j,j+\beta -1)\notin \lambda \). Then, by (2.2), we know thatHowever, since \(h_{(j,j+\beta -1)} \le 1-i\), it has to be$$\begin{aligned} h_{(j,j+\beta -1)}=0~~ \text {or}~~+\infty . \end{aligned}$$and then, by (2.2), it has to be \(\beta =1\), i.e., \((j,j) \notin \lambda \). Then, for any nonnegative integer$$\begin{aligned} h_{(j,j+\beta -1)} =0, \end{aligned}$$
*x*, \((j+x+\alpha -1, j+x) \notin \lambda \). As seen above in the second case in Case 1, we get the desired inequality.

*z*between

*c*and

*d*. Further explanations on nodes (

*j*,

*j*) and \((j',j')\) will be given in Remark 2.7.

### Lemma 2.3

*x*with \(0 \le x \le \alpha -1\),

### Proof

*z*between

*a*and

*b*. Further explanations on nodes (

*j*,

*j*) and \((j',j')\) will be given in Remark 2.7.

### 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 (*j*, *j*), its sign will be determined by the hook differences at the nodes that are in the hook of (*j*, *j*) and on the diagonals \(1-\beta \) or \(\alpha -1\).

### Definition 2.5

*j*, suppose that a node \((j,j) \in \lambda \). Then the node (

*j*,

*j*) 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.

*j*,

*j*) cannot be \((K,i,\alpha , \beta )\)-negative and positive at the same time. Also, we see that the node (

*j*,

*j*) is \((K,i,\alpha , \beta )\)-neutral if and only if

*p*,

*n*and

*e*in the boxes stand for positive, negative and neutral, respectively.

### Remark 2.6

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

### Remark 2.7

- (i)
Suppose that a node (

*j*,*j*) 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 (*j*,*j*) is negative, then any nodes between (*j*,*j*) and \((j',j')\) are negative or neutral. - (ii)
Suppose that a node (

*j*,*j*) 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 (*j*,*j*) is positive, then any nodes between (*j*,*j*) 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

*x*such that \(\lambda _x\ge x\). Then the Frobenius symbol of \(\lambda \) is defined as

*j*th 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.

### 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).

- (i)To check more easily if a node is \((K,i,\alpha ,\beta )\)-negative, we write the Frobenius symbol as follows: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$$\begin{aligned} \begin{array}{lllllll} &{} &{} a_1 &{} \cdots &{} a_{\delta -\beta +1} &{} \cdots &{} a_{\delta }\\ b_1&{} \cdots &{} b_{\beta }&{} \cdots &{} b_{\delta }. \end{array} \end{aligned}$$then the node ($$\begin{aligned} a_{j}-b_{\beta +j-1} \le 2\beta -1-i, \end{aligned}$$(2.6)
*j*,*j*) is negative. Let us turn to nodes (*j*,*j*) 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 (*j*,*j*) 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.

- (ii)Similarly, to check if a node is \((K,i,\alpha ,\beta )\)-positive, we write the Frobenius symbol as follows.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$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} a_1 &{} \cdots &{} a_{\alpha } &{} \cdots &{} a_{\delta }\\ &{} &{} b_1&{} \cdots &{} b_{\delta -\alpha +1}&{} \cdots &{} b_{\delta } \end{array} \end{aligned}$$then the node ($$\begin{aligned} a_{\alpha +j-1}-b_j\ge K-i+1-2\alpha , \end{aligned}$$(2.7)
*j*,*j*) is positive. Similarly, we can show that nodes (*j*,*j*) 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 \(j=\delta -\alpha +1\). - (iii)Adopting the notation for cylindric partitions [14], we may combine the expressions in (i) and (ii) above as follows: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).$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} &{} &{} &{} &{} a_1 &{} \cdots &{} \cdots &{} \cdots &{} a_{\delta }\\ &{} &{} b_1&{} \cdots &{} b_{\beta }&{} \cdots &{} \cdots &{} b_{\delta } \\ a_1 &{} \cdots &{} a_{\alpha } &{} \cdots &{} \cdots &{} a_{\delta }. \end{array} \end{aligned}$$

### Example 2.10

*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

*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

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

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

## 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

*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)Here we note that \(d_m^t\) can be given explicitly as follows:
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.

which is indeed a partition since \(\pi _j\ge \pi _{j+1}\), \(\pi _j'\ge \pi '_{j+1}\) for any \(j\ge 1\), and$$\begin{aligned} d^t_m(\pi )=(\pi '_1+m-1, \ldots , \pi '_t+m-1, \pi _1-t, \pi _2-t, \ldots ), \end{aligned}$$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$$\begin{aligned} \pi '_t+m-1\ge \pi _1-t \end{aligned}$$So,$$\begin{aligned} h_{(1,2)}=-1 \le m=-1. \end{aligned}$$$$\begin{aligned} d^{2}_{-1} (\pi )= (2, 2, 2, 2). \end{aligned}$$ - Case 2: \(m>0\). Let \(\pi \) be a partition such that \((t,1)\in \pi \) andThen,$$\begin{aligned} h_{(t,1)} \ge m \quad (\text {if and only if } \quad \pi _t-\pi '_{1} \ge 1-t + m). \end{aligned}$$(3.2)For example, let \(t=2, m=1\) and \(\pi =(4,4,2,2)\). We have$$\begin{aligned} d^t_{m}(\pi )= \left( d^t_{-m}(\pi ')\right) '. \end{aligned}$$By the definition,$$\begin{aligned} h_{(2,1)}=1 \ge m=1. \end{aligned}$$$$\begin{aligned} d^{2}_{1} (\pi )= (4 , 4). \end{aligned}$$

### Remark 3.1

### 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

*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

### 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.

### 3.2 The Shift Map

*u*, a shift map \(s_u\) is defined as follows [15]:

## 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

*x*th part of \(\lambda \) as

*x*in the concatenated diagram of \(\lambda \) (see Fig. 9). We now define the hook difference at a node (

*x*,

*y*) of \(\lambda \) as

### Lemma 4.2

### Proof

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

*f*,

*g*,

*h*with \(g\ge 1\), \(f\le 2g-1\), and \(h\ge f\), let \(\lambda =\nu \sigma \) with

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

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

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

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

- (a)
\(\mu \) is a partition.

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

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

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

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

### Proof

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.

(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 \(\mu \).

### Lemma 4.4

*f*,

*g*,

*h*with \(g -f \ge 1\), \(f \le 2g- 1\), \(h\le f\), let \(\lambda =\nu \sigma \) with

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

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

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

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

- (a)
\(\mu \) is a partition.

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

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

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

- (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

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

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

### Proof

*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

*w*.

(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

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

- (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.

## 5 Bijections

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

- S1.
there are no dotted blocks, or

- S2.
there are consecutive dotted blocks starting from the first non-neutral block, or

- S3.
there are consecutive dotted blocks starting from the second non-neutral block.

*EPNPN*, then the following are all the dotted blocks:

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

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

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

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

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.

*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

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 \(\psi ^{\alpha ,\beta }_{m}(\lambda )=\Gamma _{m+1}\).

*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 \(\Gamma _{v+1}\) asIn Lemma 4.3, set \(f=2-i\), \(g=wK+1\), \(h=K-i-1\), and \(\nu =D_{v+1}\), \(\sigma =\Gamma _{v}\). Clearly$$\begin{aligned} \Gamma _{v+1}=s_{-i-wK}(D_{v+1})\, (d^{\beta }_{1-i-(v-1)K}(\Gamma _{v}))'. \end{aligned}$$
*f*,*g*,*h*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),Thus, by the induction hypothesis (5.4),$$\begin{aligned} h_{(t+1,t+\beta )}(D_{v+1}\Gamma _v) =h_{(1,\beta )}(\Gamma _{v}). \end{aligned}$$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$$\begin{aligned} h_{(t+1,t+\beta )}(D_{v+1}\Gamma _v) =h_{(1,\beta )}(\Gamma _{v})\le 1-i-2wK=f-2g+1, \end{aligned}$$*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,On the other hand, since$$\begin{aligned} h_{(j,j+\beta -1)}(D_{v+1}\Gamma _v) = h_{(j,j+\beta -1)}(D_{v+1}D_v\Gamma _{v-1}). \end{aligned}$$(5.6)and \(D_v\) is negative, i.e., there are at least \(\beta \) columns, the hook difference at the node$$\begin{aligned} \Gamma _v=s_{wK}(D_{v})\, \big (d^{\alpha }_{-1+i+(v-2)K}(\Gamma _{v-1})\big )', \end{aligned}$$*b*is affected only by the shift map \(s_{wK}\) that is applied to \(D_v\), namely,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,$$\begin{aligned} h_{(j,j+\beta -1)}(D_{v+1}\Gamma _v)&=h_{(j,j+\beta -1)}(D_{v+1}D_{v}\Gamma _{v-1})-wK \nonumber \\&=h_{(j,j+\beta -1)}(D_{v+1}D_{v}\Gamma _{v-1})- (g-1). \end{aligned}$$(5.7)Therefore, by (5.6), (5.7), and (5.8),$$\begin{aligned} h_{(j,j+\beta -1)}(D_{v+1}D_v\Gamma _{v-1}) \ge 2-i =f. \end{aligned}$$(5.8)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.,$$\begin{aligned} h_{(j,j+\beta -1)}(D_{v+1}\Gamma _v) \ge {\left\{ \begin{array}{ll} f, &{} \hbox { for}\ 1\le j\le t-\beta +1,\\ f- g+1, &{} \hbox { for}\ t-\beta +2\le j\le t, \end{array}\right. } \end{aligned}$$Since \(D_{v+1}D_{v}\) forms a Frobenius symbol, we have$$\begin{aligned} (D_{v+1} | D_{v})=\left( \begin{array}{cc|cc} \cdots &{} x_1 &{} z_1 &{} \cdots \\ \cdots &{} x_2 &{} z_2 &{} \cdots \end{array}\right) . \end{aligned}$$By Lemma 4.6 (a), we know that \(z_1 \ge wK+\beta -1\). Since$$\begin{aligned} x_1>z_1, \quad x_2>z_2. \end{aligned}$$the first column of \(\Gamma _{v}\) is \(\begin{array}{l} z_1-wK\\ z_2+wK \end{array}.\) Thus, we have$$\begin{aligned} \Gamma _{v}= {\left\{ \begin{array}{ll} D_1, &{} \hbox { for}\ v=1, \\ s_{wK}(D_{v}) (d^{\alpha }_{-1+i +(v-2)K}(\Gamma _{v-1}))', &{} \hbox { for}\ v>2, \end{array}\right. } \end{aligned}$$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$$\begin{aligned} x_1>(z_1-wK)+ (wK+1)-1&\text { and } z_1-wK\ge \beta -1,\\ x_2>(z_2+wK)- (wK+1)+1&\text { and } x_2 >z_2 \ge 0, \end{aligned}$$Thus, by Statement (c) of Lemma 4.3,$$\begin{aligned} h_{(\alpha ,1)}(D_{v+1}) \ge K-i-1=h. \end{aligned}$$which verifies (5.5).$$\begin{aligned} h_{(\alpha ,1)}(\Gamma _{v+1})\ge h-2f+2g+2=-1+i+(2w+1)K, \end{aligned}$$ - Case 2: Suppose \(v=2w\) for some \(w\ge 1\). Then we can write \(\Gamma _{v+1}\) asIn Lemma 4.4, set \(f=K-i-2\), \(g=wK-1\), \(h=1-i\), and \(\nu =D_{v+1}\), \(\sigma =\Gamma _{v}\). Clearly,$$\begin{aligned} \Gamma _{v+1}=s_{wK}(D_{v+1})\,(d^{\alpha }_{-1+i+(v-1)K}(\Gamma _{v}))'. \end{aligned}$$
*f*,*g*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 thatThus, by the induction hypothesis (5.5),$$\begin{aligned} h_{(t+\alpha ,t+1)}(D_{v+1}\Gamma _v) =h_{(\alpha ,1)}(\Gamma _{v}). \end{aligned}$$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$$\begin{aligned} h_{(t+\alpha ,t+1)}(D_{v+1}\Gamma _v)&=h_{(\alpha ,1)}(\Gamma _{v})\ge -1+i+(2w-1)K\\&=-f+2g-1, \end{aligned}$$*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,On the other hand, since$$\begin{aligned} h_{(j+\alpha -1,j)}(D_{v+1}\Gamma _v) = h_{(j+\alpha -1,j)}(D_{v+1}D_v\Gamma _{v-1}). \end{aligned}$$(5.9)and \(D_v\) is positive, i.e., there are at least \(\alpha \) columns, the hook difference at the node$$\begin{aligned} \Gamma _v=s_{-i-(w-1)K}(D_{v})\, (d^{\beta }_{1-i-(v-2)K}(\Gamma _{v-1}))', \end{aligned}$$*d*is affected only by the shift map \(s_{-i-(w-1)K}\) that is applied to \(D_v\), namely,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,$$\begin{aligned} h_{(j+\alpha -1,j)}(D_{v+1}\Gamma _v)&=h_{(j+\alpha -1,j)}(D_{v+1}D_{v}\Gamma _{v-1})+i+(w-1)K \nonumber \\&=h_{(j+\alpha -1,j)}(D_{v+1}D_{v}\Gamma _{v-1})-f+g-1. \end{aligned}$$(5.10)Therefore, by (5.9), (5.10), and (5.11),$$\begin{aligned} h_{(j+\alpha -1,j)}(D_{v+1}D_v\Gamma _{v-1}) \le K-i-2 =f. \end{aligned}$$(5.11)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).$$\begin{aligned} h_{(j+\alpha -1,j) }(D_{v+1}\Gamma _v) \le {\left\{ \begin{array}{ll} f, &{} \hbox { for}\ 1\le j\le t-\alpha +1,\\ g-1, &{} \hbox { for}\ t-\alpha +2\le j\le t, \end{array}\right. } \end{aligned}$$

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.

*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

\(\Gamma _1 = D_1 = \left( \begin{array}{cc} 1 &{} 0 \\ 1 &{} 0 \end{array} \right) \),

## 6 Results

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

*m*th block, then the

*m*th 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,

*n*with an overlined entry in its

*m*th anchor, which is defined before Theorem 1.2 in Introduction.

### Theorem 6.1

*p*(

*N*) denotes the number of ordinary partitions of

*N*with \(p(0)=1\) and \(p(N)=0\) for \(N<0\).

### Remark 6.2

- (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 havefor \(m\ge 1\) and \(n\ge 0\).$$\begin{aligned}&{\overline{Q}}_{K,i, \alpha , \beta } (m, n)\\&~=p\left( n-\left( K \left\lceil \frac{m}{2}\right\rceil ^2 - i\left\lceil \frac{m}{2}\right\rceil \right) \alpha - \left( K \left\lfloor \frac{m}{2}\right\rfloor ^2 + (K-i)\left\lfloor \frac{m}{2}\right\rfloor \right) \beta \right) \\&\quad +p\left( n-\left( K \left\lceil \frac{-m}{2}\right\rceil ^2 - i\left\lceil \frac{-m}{2}\right\rceil \right) \alpha - \left( K \left\lfloor \frac{-m}{2}\right\rfloor ^2 + (K-i)\left\lfloor \frac{-m}{2}\right\rfloor \right) \beta \right) \end{aligned}$$
- (ii)Each ordinary partition of
*n*can be regarded as a \((K,i,\alpha ,\beta )\)-singular overpartition without any overlined entries. Thus$$\begin{aligned} {\overline{Q}}_{K,i,\alpha ,\beta }(0,n)=p(n). \end{aligned}$$

### Theorem 6.3

### Proof

### 6.1 Proof of 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

## 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.

*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

*m*th anchor, it is the same as the number of partitions of

*n*with at least

*m*signed blocks. By the sieving method, we have

## Notes

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