A New Approach to Integer Partitions

Abstract

In this work we define a new set of integer partition, based on a lattice path in \({\mathbb {Z}}^2\) connecting the line \(x+y=n\) to the origin, which is determined by the two-line matrix representation given for different sets of partitions of n. The new partitions have only distinct odd parts with some particular restrictions. This process of getting new partitions, which has been called the Path Procedure, is applied to unrestricted partitions, partitions counted by the 1st and 2nd Rogers–Ramanujan Identities, and those generated by the Mock Theta Function \(T_1^*(q)=\sum _{n=0}^{\infty }\dfrac{q^{n(n+1)}(-q^2,q^2)_n}{(q,q^2)_{n+1}}\).

Introduction

Two-line matrices representing integer partitions appeared, for example, in Frobenius (1900) and Andrews (1984), where the Frobenius representation and F-partitions were defined. In a different way, two-line matrix representations for partitions were introduced in Santos et al. (2011) and also treated in Brietzke et al. (2010, 2011, 2013), Bagatini et al. (2017), and Matte (2017). One of the advantages of these representations over the Frobenius’ one is that, in the case of unrestricted partitions, the conjugated partition is completely determined in the entries of the second line. The present work deals with a new approach to these matrix representations, whose similar meaning has already been given by Alegri et al. Alegri et al. (2011) for matrices representing plane partitions. As it has been pointed out in Santos et al. (2011), each two-line matrix representation for the partitions of n can be associated to a lattice path through the Cartesian plane, connecting the origin (0, 0) to the line \(x+y=n\). Although a description of a possible path is rapidly done there, in this paper we explore this approach more deeply.

Since every matrix described in Santos et al. (2011) is of type

$$\begin{aligned} \left( \begin{array}{ccccc} c_1 &{} c_2 &{} c_3 &{} \cdots &{} c_s \\ d_1 &{} d_2 &{} d_3 &{} \cdots &{} d_s \end{array} \right) , \end{aligned}$$

and its entries sum n, we consider the entries of each matrix as a guidance for a lattice path in \({\mathbb {Z}}^2\) from de line \(x+y=n\) to the origin (0, 0). In each matrix the entries \(c_i\) of the first row determine the moves along the y-axis and the entries \(d_i\) of the second row determine the moves along the x-axis. Starting at the point \(P=(\sum _{i=1}^{s} d_i,\sum _{i=1}^{s} c_i)\), the path consists of shifting \(c_i\) units down followed by \(d_i\) units to the left, for i from s to 1, where s is the number of columns of the matrix.

In the following pages we describe the lattice paths induced by the two-line matrix representations and associate different sets of integer partitions to them. In particular, we deal with the matrix representations for unrestricted partitions, partitions counted by the 1st and 2nd Rogers–Ramanujan Identities, and those generated by the Mock Theta Function \(T_1^*(q)=\sum _{n=0}^{\infty }\dfrac{q^{n(n+1)}(-q^2,q^2)_n}{(q,q^2)_{n+1}}\).

Each path defined above is then associated to a partition of another integer, which turns out to have only distinct odd parts, and these partitions are the object of study of the present work. The procedure of getting partitions from a two-line matrix, which we call the Path Procedure, can be applied to any set of integer partitions whose two-line matrix representation is known. And as mock theta functions can also be seen as generating functions for certain partitions, the Path Procedure can be applied to them too, provided that their two-line matrix representation is stated. Since there are a lot of known matrix representations, although not presented in the present pages, this in any way means that the possibilities of this approach are exhausted; much more achievement is expected.

The Path Procedure

Let us consider the unrestricted integer partitions, which have at least three different two-line matrix representation. We choose the following one.

Theorem 2.1

(Theorem 4.1, Santos et al. 2011) The number of unrestricted partitions of n is equal to the number of two-line matrices of the form

$$\begin{aligned} \left( \begin{array}{ccccc} c_1 &{} c_2 &{} c_3 &{} \cdots &{} c_s \\ d_1 &{} d_2 &{} d_3 &{} \cdots &{} d_s \end{array} \right) , \end{aligned}$$
(1)

where \(c_s=0\), \(c_{t}=c_{t+1}+d_{t+1}\), and the sum of all entries is equal to n.

Observe that for every matrix the sum of the entries of each column gives the respective part of the original partition. That is, the \(k^{th}\) part of the associated partition of n is equal to \(c_k+d_k\). For \(n=6\) we have the following.

Example 2.2

For \(n=6\) we have \(p(6)=11\), and so there are 11 matrices satisfying Theorem 2.1, shown in Table 1.

Table 1 Table for Example 2.2

We associate each matrix of Theorem 2.1 (and thus each partition of n) to a path built through the Cartesian plane, connecting the point \(P=(\sum _{i=1}^{s} d_i,\sum _{i=1}^{s} c_i)\) in the line \(x+y=n\) to the origin (0, 0). We choose the second line of the matrix to be associated to the x-axis, and the first line to be associated to the y-axis.

The path consists of shifting \(c_s\) units down, \(d_s\) units to the left, then \(c_{s-1}\) units down, \(d_{s-1}\) units to the left, and so on, ending with \(d_1\) units to the left. So we create a path which connects the following points:

$$\begin{aligned} P= & {} \bigg (\sum _{i=1}^{s} d_i,\sum _{i=1}^{s} c_i\bigg )\rightarrow \bigg (\sum _{i=1}^{s} d_i,\sum _{i=1}^{s-1} c_i\bigg )\rightarrow \bigg (\sum _{i=1}^{s-1} d_i,\sum _{i=1}^{s-1} c_i\bigg )\rightarrow \bigg (\sum _{i=1}^{s-1} d_i,\sum _{i=1}^{s-2} c_i\bigg ) \\&\rightarrow \bigg (\sum _{i=1}^{s-2} d_i,\sum _{i=1}^{s-2} c_i\bigg )\rightarrow \ldots \rightarrow (d_1+d_2, c_1+c_2)\rightarrow (d_1+d_2,c_1)\\&\rightarrow (d_1,c_1)\rightarrow (d_1,0)\rightarrow (0,0). \end{aligned}$$

Example 2.3

For \(n=6\) we take the matrix

$$\begin{aligned} M=\left( \begin{array}{cccc} 2&{}1&{}1&{}0 \\ 0&{}1&{}0&{}1 \end{array} \right) \end{aligned}$$

associated to the partition (2, 2, 1, 1). The path associated to M connects the points (2, 4), (1, 4), (1, 3), (1, 2), (0, 2), and (0, 0), as shown in Fig. 1.

Fig. 1
figure1

Illustration for Example 2.3

Remark 2.4

Note that \(\big (\sum _{i=1}^{s} d_i,\sum _{i=1}^{s-1} c_i\big )=P\), since \(c_s=0\).

Remark 2.5

According to the conditions satisfied by the entries of each matrix of type (1), note that every down move is necessarily at least as large as the previous left move.

Now we reflect the path through the line \(x+y=n\) and create the parts of a new partition by taking hooks of the following sizes.

$$\begin{aligned}&2(n-d_1)-1; \\&2(n-d_1-1)-1; \\&\cdots \\&2(n-d_1-(c_1-1))-1; \\&2(n-d_1-d_2-c_1)-1; \\&2(n-d_1-d_2-c_1-1)-1; \\&\cdots \\&2(n-d_1-d_2-c_1-(c_2-1))-1; \\&\cdots \\&2(n-d_1-\cdots -d_k-c_1-\ldots -c_{k-1})-1; \\&2(n-d_1-\cdots -d_k-c_1-\ldots -c_{k-1}-1)-1; \\&\cdots \\&2(n-d_1-\cdots -d_k-c_1-\ldots -c_{k-1}-(c_k-1))-1; \\&\cdots \\&2(n-d_1-\cdots -d_s-c_1-\ldots -c_{s-1}+1)-1. \end{aligned}$$

Based on the construction above, we get a partition of some integer m into distinct odd parts greater than 1 and less than or equal to \(2n-1\), since the matrix representation of an original unrestricted partition of n has entry \(d_1\ge 0\). Therefore, \(m\le n^2-1\).

Example 2.6

For the matrix

$$\begin{aligned} M=\left( \begin{array}{cccc} 2&{}1&{}1&{}0 \\ 0&{}1&{}0&{}1 \end{array} \right) \end{aligned}$$

associated to the partition (2, 2, 1, 1) of \(n=6\), the hooks given by the reflection of the path through the line \(x+y=6\) provide the parts 11, 9, 5,  and 3, that is, the partition \(\mu =(11,9,5,3)\) of \(m=28\). Figure 2 below helps to understand the process.

Fig. 2
figure2

Illustration for Example 2.6

Example 2.7

By considering all the matrices in Example 2.2 we get the partitions contained in Table 2.

Table 2 Table for Example 2.7

It is clear that each matrix associated to a different partition of n generates a different path from the line \(x+y=n\) to the origin (0, 0), and therefore a different partition into distinct odd parts. However, there are matrices associated to partitions of different integers n that induce different paths but same hooks, generating the same partition into distinct odd parts. This fact is illustrated in the next example.

Example 2.8

Let us take the partition (4, 1, 1) of 6 and the partition (2, 1, 1) of 4. The matrices associated to them are, respectively,

$$\begin{aligned} \left( \begin{array}{ccc} 1&{}1&{}0 \\ 3&{}0&{}1 \end{array} \right) \text{ and } \left( \begin{array}{ccc} 1&{}1&{}0 \\ 1&{}0&{}1 \end{array} \right) . \end{aligned}$$

The paths that these matrices origin are different, although both paths induce the same hooks and, therefore, the same partition into distinct odd parts, as shown in Fig. 3.

Fig. 3
figure3

Illustration for Example 2.8

Definition 2.9

We call the process described above the Path Procedure. More precisely, from now on, we use the denomination Path Procedure when referring to the process of building partitions into distinct odd parts from the two-line matrix representation of a specific set of integer partitions. The construction consists of connecting the point \(P=(\sum _{i=1}^{s} d_i,\sum _{i=1}^{s} c_i)\) in line \(x+y=n\) to the origin (0, 0) by shifting successively \(c_i\) units down and \(d_i\) units to the left, for i from s to 1, reflecting the path through the line \(x+y=n\), and taking hooks of odd sizes which will constitute the parts of the new partition.

The previous considerations motivate questions like:

Question 2.10

Which integers are being partitioned into distinct odd parts by the process described above?

Question 2.11

Given a partition into distinct odd parts, is it generated by an unrestricted partition, according to our construction?

Question 2.12

In case of an affirmative answer to Question 2.11, how to recover the matrix which induced the given partition into distinct odd parts?

We intend to answer these questions in the following pages.

Recovering a Matrix

We start with some general observations which will lead us to a map from some specific partitions into distinct odd parts to the matrix representation from Theorem 2.1.

Remark 3.1

Given a partition into distinct odd parts \(\lambda =(2\lambda _i+1, 2\lambda _{i-1}+1, \ldots , 2\lambda _2+1, 2\lambda _1+1)\), with \(\lambda _j> \lambda _{j-1}\), observe that there are \(\lambda _j-\lambda _{j-1}-1\) distinct odd integers between \(2\lambda _j+1\) and \(2\lambda _{j-1}+1\).

Let \(\lambda \) be a partition obtained from the matrix representation of an unrestricted partition of n, and call its smallest part \(2\lambda _1+1\). According to the Path Procedure, the hook generated by the path from the line \(x+y=n\) to the origin (0, 0), which gave origin to the smallest part \(2\lambda _1+1\), was obtained from a left move of \(\lambda _1\) units. So, \(c_s=0\) and \(d_s=\lambda _1\). Necessarily, \(c_{s-1}=\lambda _1\), which means that a down move of \(\lambda _1\) units generates a sequence of \(\lambda _1\) consecutive odd parts, which are \(2\lambda _1+1, 2(\lambda _1+1)+1, 2(\lambda _1+2)+1, \ldots , 2(2\lambda _1-1)+1\).

Now there are two possibilities:

(a):

If \(2(2\lambda _1)+1\) is a part of the partition \(\lambda \), it means that it was generated by a down move, with no left move before it. In this case, \(d_{s-1}=0\) and \(c_{s-2}=\lambda _1\), which means that again there is a sequence of \(\lambda _1\) consecutive odd parts \(2(2\lambda _1)+1, 2(2\lambda _1+1)+1, \ldots , 2(3\lambda _1-1)-1\).

(b):

If \(2(2\lambda _1)+1\) is not a part of the partition \(\lambda \), then a left move is allowed. In this case, let us call \(2\lambda _2+1\) the first part that appears in \(\lambda \) after \(2(2\lambda _1-1)+1\). So \(d_{s-1}=\lambda _2-(2\lambda _1-1)-1=\lambda _2-2\lambda _1\) and \(c_{s-2}=\lambda _2-\lambda _1\).

By repeating an analogous argument until the last part of the partition \(\lambda \), we obtain the values of \(c_j\) for all \(j\ge 1\), and \(d_j\) for all \(j\ge 2\). The value of \(d_1\) is the size of the last left move of the path. Observe that it does not generate any odd part, and so it can be as large as we want. In other words, the entry \(d_1\) of the matrix does not affect the size of the partition into distinct odd parts; it only affects the size of n in the original unrestricted partition.

Now, having a matrix representation which originated the partition into distinct odd parts (which turns out to be precisely the representation given by Theorem 2.1), by summing its columns we get the original unrestricted partition of n that induced the partition into distinct odd parts.

Example 3.2

Let us take the partition (11, 9, 5, 3). By considering all the positive odd integers we see that, before the part 3, the odd integer 1 is omitted. This means that the path between \(x+y=n\) and (0, 0) starts with a left move of size 1, which implies \(d_s=1\). Consequently, \(c_{s-1}=1\). This implies a down move of size 1, which generates the part 3.

As the next part is the consecutive odd number 5, we have one more down move of size 1, with no left move in between. This means that \(d_{s-1}=0\) and \(c_{s-2}=1\).

After that, the number 7 is omitted because of a left move of size 1, which means that \(d_{s-2}=1\). Necessarily \(c_{s-3}=2\), and so we have a down move of size 2, which generates the last two parts, 11 and 9.

By this construction, \(s-3=1\) and so \(s=4\). The entry \(d_1\) has the size of the last left move necessary to reach the origin (0, 0).

As illustrated in Example 2.8, the same partition into distinct odd parts can be generated by different matrices, i.e., by different unrestricted partitions. The difference occurs in the entry \(d_1\), whose size depends on the size of n. In other words, the first line of the matrix associated to some partition of n expresses the size of the sequences of consecutive odd parts less than or equal to \(2n-1\) that appear in the partition. The second line expresses the size of the sequence of consecutive odd numbers that are not parts of the partition.

By recursion it is easy to note that \(c_i=\sum _{j=i+1}^{s}d_j\), and so we can conclude that, given an increasing finite sequence of distinct odd integers, it is a partition generated by a matrix, according to Theorem 2.1, if the size of any subsequence of consecutive odd numbers is either exactly the number of smaller odd integers that were omitted before the subsequence started or a multiple of it. Note that the part 1 does never appear but the first part may be any odd integer greater than or equal to 3.

The Path Procedure Applied to Unrestricted Partitions

In this section we set some results obtained from the Path Procedure applied to unrestricted partitions of n.

Recalling Definition 2.9, the new parts obtained by the Path Procedure constitute a partition of some integer \(m<n^2\) into distinct odd parts, whose size of any subsequence of consecutive odd numbers that are parts of the partition is either exactly the number of smaller odd integers that were omitted before the subsequence started or a multiple of it.

For each value of n we count how many times each integer m appears in the construction described above. This data is organized in squares of size \(n\times n\) (or tables with \(n^2\) cells), as we can see in Figs. 4 and 5 below.

Fig. 4
figure4

\(n\times n\) squares for \(n=4 \text{ and } 5\)

Fig. 5
figure5

\(n\times n\) square for \(n=21\)

The figures illustrate the distribution of frequencies of partitions of m in squares of size \(n\times n\), induced by the partitions of n. Each cell contains how many partitions of m (m is indicated in the right down side of the cell) are generated by the Path Procedure applied to partitions of n.

Another representation for the distribution of frequencies is presented in Fig. 6. We organize in columns the same frequencies contained in the cells of the figures above. We show the case for \(n=8\).

Fig. 6
figure6

\(n\times n\) square for \(n=8\) and its representation by columns

Note that the column for \(m=48\) has height 2. This happens because the number 48 is generated by two different partitions into distinct odd parts greater than 1 and less than or equal to \(2\cdot 8-1=15\). That is, the partitions (13, 11, 9, 7, 5, 3) and (15, 13, 11, 9), coming from the original partitions of 8, (2, 1, 1, 1, 1, 1, 1) and (4, 4), respectively.

Observe that, as n gets larger, the number of partitions of \(m<n^2\) increases quickly. That is, the number of partitions of \(m<n^2\) grows faster than n. So, the frequency columns can get very high. However, these frequencies clearly cannot grow indefinitely; as n gets larger, new distinct odd parts are allowed but they will not be used in every partition. At some point, every column reaches a limited height.

By considering all the remarks and restrictions in the process of getting new partitions, the Path Procedure motivates the following definition.

Definition 4.1

Let \(P_{od}(m)\) be the set of partitions of m into distinct odd parts greater than 1 whose size of any subsequence of consecutive odd integers is either exactly the number of smaller odd integers that were omitted before the subsequence started or a multiple of it. Also, \(\big \vert P_{od}(m)\big \vert =p_{od}(m)\).

Remark 4.2

If no odd integer is omitted after some subsequence of parts, we assume the number of omitted parts is zero, and the size of the following subsequence of parts will have the same size of the previous one. Note that the integer 1 is never a part, so the first subsequence of omitted parts has at least size 1.

As an example, we take the cell of \(m=232\) in Fig. 5, and so the greatest part allowed in any of its partitions is \(2\cdot 21-1=41\) (this is important for determining the value of the entry \(d_1\) of the associated matrices). As this cell has the number 5 in it, this means \(p_{od}(m)=5\).

Example 4.3

For \(n=21\) and \(m=232\) we have \(p_{od}(m)=5\), as it follows (Table 3).

Table 3 Table for Example 4.3

With all the considerations made until now, we can assure that both sets of unrestricted partitions of n and of partitions defined by Definition 4.1 have the same cardinality.

In the following pages we set some results that characterize the numbers contained in squares like the ones in Figs. 4, 5, and 6. We begin with a characterization of the numbers m that appear with frequency zero in our squares.

Proposition 4.4

For all \(n\ge 0\) we have

$$\begin{aligned} p_{od}(4n+1)=0=p_{od}(4n+2). \end{aligned}$$

Proof

We prove that \(p_{od}(4n+1)=0\). The other equality has an analogous proof.

Let us suppose, by absurd, that \(p_{od}(4n+1)\ne 0\). So, we would like to partition \(4n+1\) into distinct odd parts greater than or equal to 3. For these parts to sum \(4n+1\), there are the following possibilities:

  • (i) a number \(\equiv 1 \pmod 4\) of parts, all of them \(\equiv 1 \pmod 4\);

  • (ii) a number \(\equiv 3 \pmod 4\) of parts, all of them \(\equiv 3 \pmod 4\);

  • (iii) j parts \(\equiv 1 \pmod 4\) and a number \(\equiv j-1\pmod 4\) of parts \(\equiv 3 \pmod 4\);

  • (iv) j parts \(\equiv 3 \pmod 4\) and a number \(\equiv j-3\pmod 4\) of parts \(\equiv 1 \pmod 4\).

Note that there are no more possible configurations for the parts, since their sums would be even or \(\equiv 3 \pmod 4\).

Let us begin with case (i), supposing we could partition \(4n+1\) as

$$\begin{aligned} \lambda =\big (2(2k_j+1)-1, 2(2k_{j-1}+1)-1, \ldots , 2(2k_2+1)-1, 2(2k_1+1)-1\big ), \end{aligned}$$

with \(j\equiv 1 \pmod 4\) and \(k_i>k_{i-1}.\)

In this case, observe that, as 1 cannot be a part of \(\lambda \), the smallest odd part that can appear is 5. So, \(d_s\ge 2\) and then \(c_{s-1}=c_s+d_s=d_s\ge 2\). However, every \(c_i\) should be 1, since there are no consecutive odd parts in \(\lambda \). As this is a contradiction, case (i) can never occur.

With an analogous argument, case (ii) is also not possible: if we could partition \(4n+1\) as

$$\begin{aligned} \lambda =\big (2(2k_j+2)-1, 2(2k_{j-2}+2)-1, \ldots , 2(2k_2+2)-1, 2(2k_1+2)-1\big ), \end{aligned}$$

with \(j\equiv 3 \pmod 4\) and \(k_i>k_{i-1}\), observe that again there would not be consecutive odd parts in \(\lambda \), which means \(d_i\ne 0\; \forall i\) and \(c_i=1 \; \forall i\ne s\). But then we get \(c_{s-1}\ge 2\), and so this case cannot occur either.

In case (iii) observe that in a partition with j parts \(\equiv 1 \pmod 4\) and a number \(\equiv j-1\pmod 4\) of parts \(\equiv 3 \pmod 4\), essentially two configurations are possible: either all the parts are non-consecutive odd integers, or there is at least a subsequence of two consecutive odd parts.

In the first configuration, with no consecutive odd parts, the same argument used in cases (i) and (ii) is valid: every \(c_i \; \forall i\ne s\) should be 1 when they are actually not. So this configuration is not possible.

If there is a subsequence of consecutive odd parts, let us say it has size r and suppose it is the first subsequence of consecutive odd parts of the partition.

If this subsequence does not contain the smallest parts of the partition, again we have the problem of existing a t such that \(c_t\) must be 1 when it is actually not. So, let us suppose the r consecutive odd parts are the smallest parts of the partition, saying \(2k_r-1, 2(k_r-1)-1,\ldots ,2(k_r-r+1)-1\). In this situation, \(c_s=0\), \(d_s=k_r-r\), and \(c_{s-1}=k_r-r\).

If \(k_r-r> r\) we have a contradiction. If \(k_r-r< r\), then \(d_{s-1}=0\) and \(c_{s-2}=k_r-r\). By repeating this argument (which has an end, since the sequence \(2k_r-1, 2(k_r-1)-1,\ldots ,2(k_r-r+1)-1\) is finite), we will find a t such that \(c_t\) must be 1 when it is actually not. If \(k_r-r=r\), then \(d_{s-1}>0\) and \(c_{s-2}\ge r\), and we may start a new subsequence of consecutive odd parts. By analysing the parity of \(k_r-r\), r, and the others \(c_i\) and \(d_i\), we conclude that no configuration allows j parts \(\equiv 1 \pmod 4\) and a number \(\equiv j-1\pmod 4\) of parts \(\equiv 3 \pmod 4\). So, case (iii) does not happen either.

By noting that in case (iv) we may use the same argument as in case (iii), we conclude that \(p_{od}(4n+1)=0\).

In order to show that \(p_{od}(4n+2)= 0\), we just observe that for distinct odd parts greater than or equal to 3 to sum \(4n+2\), there are the following possibilities:

  • (i) a number \(\equiv 2 \pmod 4\) of parts, all of them \(\equiv 1 \pmod 4\);

  • (ii) a number \(\equiv 2 \pmod 4\) of parts, all of them \(\equiv 3 \pmod 4\);

  • (iii) j parts \(\equiv 1 \pmod 4\) and a number \(\equiv j-2\pmod 4\) of parts \(\equiv 3 \pmod 4\);

  • (iv) j parts \(\equiv 3 \pmod 4\) and a number \(\equiv j-2\pmod 4\) of parts \(\equiv 1 \pmod 4\).

The arguments in each case are the same we have already used and we omit them. So, \(p_{od}(4n+1)=0=p_{od}(4n+2).\)\(\square \)

In order to proof the next results we set some definitions. First, recall that the smallest odd part of any partition generated by the Path Procedure has to be greater than 1, since the matrix representation of unrestricted partitions of n has entry \(c_s=0\), which means that the path from the line \(x+y=n\) to the origin starts with a left move of size \(d_s\).

The size of the smallest part is determined by the size of \(d_s\), which we call the first missing subsequence of the partition. We understand the first missing subsequence as the first sequence of consecutive odd integers that do not appear as parts of the partition, which are \(2k_1-1, \ldots ,3,1\). Let us call \(d_s=k_1\).

After the first missing subsequence of \(k_1\) consecutive odd integers we have the first subsequence of consecutive odd parts that compose the partition. Its size is determined by the entry \(c_{s-1}\) of the matrix. As \(c_{s-1}=c_s+d_s=0+k_1=k_1\), this means that the parts of the first sequence are \(2(2k_1)-1,\ldots ,2(k_1+2)-1,2(k_1+1)-1\).

Some examples of partitions that have only the first missing subsequence and after it exactly one subsequence of consecutive odd parts are

$$\begin{aligned} (3), (7,5), (11,9,7), (15,13,11,9), \end{aligned}$$

which are, respectively, partitions of 3, 12, 27, and 48. Those partitions into odd parts have to be generated by a partition of n into two parts, since its matrix representation has two columns, which means a path with only one down move of size \(c_1=k_1\). The following result gives a general characterization of partitions like those.

Proposition 4.5

The Path Procedure applied to the unrestricted partitions of n into exactly two parts generates partitions of \(m=3k_1^2\), with \(1\le k_1\le \big \lfloor \frac{n}{2}\big \rfloor \), those being precisely all of the numbers whose partition has only the first missing subsequence and after it exactly one subsequence of consecutive odd parts.

Unrestricted partitions of n with more than two parts have a matrix representation into more than two columns, which means that each one of its entries \(c_t\) generates a different sequence of consecutive odd parts.

We call the second missing subsequence the second sequence of \(d_{s-1}=k_2\ge 0\) consecutive odd integers that do not appear as parts of the partition. They are \(2(2k_1+k_2)-1,\ldots ,2(2k_1+2)-1, 2(2k_1+1)-1\). Observe that \(k_2\) can actually be equal to 0: its size is determined by \(\lambda _{s-1}=c_{s-1}+d_{s-1}\), where \(\lambda _{s-1}\) is part of an unrestricted partition of n, and if \(\lambda _{s-1}=\lambda _s\) this means \(d_{s-1}=0\).

After the second missing subsequence of \(k_2\) consecutive odd integers we have the second subsequence of consecutive odd parts that compose the partition, determined by the size of entry \(c_{s-2}\) of the matrix. As \(c_{s-2}=c_{s-1}+d_{s-1}=k_1+k_2\), the parts of the second subsequence are \(2(3k_1+2k_2)-1,\ldots ,2(2k_1+k_2+2)-1, 2(2k_1+k_2+1)-1\). For example,

$$\begin{aligned} (5,3),(13,11,9,3),(11,9,7,5),(17,15,13,11,9,7),(25,23,21,19,17,11,9,7), \end{aligned}$$

which are, respectively, partitions of 8, 36, 32, 76, and 132. A general characterization of partitions like those is given next.

Proposition 4.6

The Path Procedure applied to the unrestricted partitions of n into exactly three parts generates partitions of \(m=8k_1^2+k_2(8k_1+3k_2)\), with \(1\le k_1\le \lfloor \frac{n}{2}\rfloor \) and \(0\le k_2\le \lfloor \frac{n-3k_1}{2}\rfloor \).

Now let us consider a partition into distinct odd parts having tmissing subsequences and tsubsequences of consecutive odd parts. We call \(k_1,k_2, \ldots ,k_t\) the sizes of the missing subsequences and, consequently, the subsequence after a missing subsequence of size \(k_i\) has size \(k_1+k_2+\cdots +k_i\). The following lemma establishes the limits for each \(k_i\).

Lemma 4.7

Theith missing subsequence of a partition into distinct odd parts, whose parts derive from the Path Procedure applied to unrestricted partitions of n, is at most

$$\begin{aligned} \dfrac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1})}{2}. \end{aligned}$$

Remark 4.8

The number of missing subsequences depends on the size of each missing subsequence. We can have from just one to even \(n-1\) missing subsequences. The first case is the one with \(k_1=\lfloor \frac{n}{2}\rfloor \); the second one is the case with \(k_1=1\) and \(k_i=0\) for \(2\le i\le n-1\).

Now we can extend our construction to a more general characterization of the numbers partitioned into distinct odd parts, whose parts derive from the Path Procedure applied to unrestricted partitions of n.

Theorem 4.9

The partitions into distinct odd parts induced by the Path Procedure applied to partitions of n are all of the form

$$\begin{aligned}&\sum _{\begin{array}{c} i=1 \\ 1\le k_1\le \frac{n}{2} \\ i>1,\, 0\le k_i\le \frac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1})}{2} \end{array}}^{t}[(t+2-i)^2-1]k_i^2 \nonumber \\&\quad +\,\sum _{\begin{array}{c} i=1 \\ 1\le k_1\le \frac{n}{2} \\ i>1,\, 0\le k_i\le \frac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1})}{2} \end{array}}^{t-1}k_{i}\sum _{j=1}^{t-i}j(2(t-i+3))k_{t-j+1}, \end{aligned}$$
(2)

where \(1\le t\le n-1\).

Proof

First of all, let us rewrite the expression (2) by expanding the sums.

$$\begin{aligned}&\sum _{\begin{array}{c} i=1 \\ 0\le k_i\le \frac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1})}{2} \end{array}}^{t}[(t+2-i)^2-1]k_i^2 \nonumber \\&\qquad +\,\sum _{\begin{array}{c} i=1 \\ 0\le k_i\le \frac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1})}{2} \end{array}}^{t-1}k_{i}\sum _{j=1}^{t-i}j(2(t-i+3))k_{t-j+1}\nonumber \\&\quad = ((t+1)^2-1)k_1^2+(t^2-1)k_2^2+\cdots +8k_{t-1}^2+3k_t^2 \nonumber \\&\qquad +\,k_1(2(t+2)k_t+2(2(t+2))k_{t-1}+\cdots +(t-2)(2(t+2))k_3\nonumber \\&\qquad +\,(t-1)(2(t+2))k_2) \nonumber \\&\qquad +\,k_2(2(t+1)k_t+2(2(t+1))k_{t-1}+\cdots +(t-2)(2(t+1))k_3) \nonumber \\&\qquad +\,\cdots \nonumber \\&\qquad +\,k_{t-3}(12k_t+24k_{t-1}+36k_{t-2}) \nonumber \\&\qquad +\,k_{t-2}(10k_t+20k_{t-1}) \nonumber \\&\qquad +\,k_{t-1}(8k_t) \end{aligned}$$
(3)

By considering a partition generated by the Path Procedure induced by an unrestricted partition of n, suppose it has t subsequences of consecutive odd parts. Let us call \(k_1\), \(k_2\), \(\ldots \), \(k_t\) the missing subsequences. So, the sizes of the subsequences of consecutive odd parts are \(k_1\), \(k_1+k_2\), \(k_1+k_2+k_3\), \(\ldots \), \(k_1+k_2+\cdots +k_t\) and the partition we are considering has the following subsequences of consecutive odd parts:

$$\begin{aligned}&2((t+1)k_1+\cdots +3k_{t-1}+2k_t)-1,\ldots , 2(tk_1+\cdots + 2k_{t-1}+k_{t}+2)-1, \\&\quad 2(tk_1+\cdots +2k_{t-1}+k_{t}+1)-1;\\&2(tk_1+\cdots +3k_{t-2}+2k_{t-1})-1,\ldots , 2((t-1)k_1+\cdots + 2k_{t-2}+k_{t-1}+2)-1, \\&\quad 2((t-1)k_1+\cdots +2k_{t-2}+k_{t-1}+1)-1; \\&\ldots \\&2(4k_1+3k_2+2k_3) -1,\ldots 2(3k_1+2k_2+k_3+2)-1,2(3k_1+2k_2+k_3+1)-1; \\&2(3k_1+2k_2) -1, \ldots , 2(2k_1+k_2+2)-1, 2(2k_1+k_2+1)-1; \end{aligned}$$

and

$$\begin{aligned} 2(2k_1)-1, \ldots , 2(k_1+2)-1, 2(k_1+1)-1. \end{aligned}$$

The sum of those parts equals

$$\begin{aligned}&\frac{(2((t+1)k_1+\cdots +3k_{t-1}+2k_t)-1}{2} \\&\qquad +\,\frac{2(tk_1+\cdots +2k_{t-1}+k_{t}+1)-1)(k_1+k_2+\cdots +k_t)}{2} \\&\qquad +\, \frac{(2(tk_1+\cdots +3k_{t-2}+2k_{t-1})-1}{2} \\&\qquad +\, \frac{2((t-1)k_1+\cdots +2k_{t-2}+k_{t-1}+1)-1)(k_1+k_2+\cdots +k_{t-1})}{2} \\&\qquad +\, \cdots \\&\qquad +\, \frac{(2(4k_1+3k_2+2k_3) -1+2(3k_1+2k_2+k_3+1)-1)(k_1+k_2+k_3)}{2} \\&\qquad +\, \frac{(2(3k_1+2k_2) -1+2(2k_1+k_2+1)-1)(k_1+k_2)}{2} \\&\qquad +\, \frac{(2(2k_1)-1+2(k_1+1)-1)(k_1)}{2} \\&\quad = ((2t+1)k_1+\cdots +5k_{t-1}+3k_t)(k_1+k_2+\cdots +k_t) \\&\qquad +\, ((2t-1)k_1+\cdots +5k_{t-2}+3k_{t-1})(k_1+k_2+\cdots +k_{t-1}) \\&\qquad +\, \cdots \\&\qquad +\, (7k_1+5k_2+3k_3)(k_1+k_2+k_3) \\&\qquad +\, (5k_1+3k_2)(k_1+k_2) \\&\qquad +\, (3k_1)(k_1). \end{aligned}$$

By rearranging the terms in the sum above we get expression (3), which proves the theorem. \(\square \)

The Path Procedure Applied to Partitions Counted by the \(1^{st}\) Rogers–Ramanujan Identity

Motivated by what we did in the previous sections, let us consider now the matrix representation for the partitions of n into 2-distinct parts, that is, partitions of n where the difference between parts is at least two. These partitions are counted by the right-hand side of the \(1^{st}\) Rogers–Ramanujan Identity,

$$\begin{aligned} p(n \vert \text{ parts } \equiv 1 \text{ or } 4\pmod 5)=p(n\vert 2\text{-distinct } \text{ parts }). \end{aligned}$$

Theorem 5.1

(Corollary 3.2, Santos et al. (2011)). The number of partitions of n where the difference between parts is at least two is equal to the number of two-line matrices of the form

$$\begin{aligned} \left( \begin{array}{ccccc} c_1 &{} c_2 &{} c_3 &{} \cdots &{} c_s \\ d_1 &{} d_2 &{} d_3 &{} \cdots &{} d_s \end{array} \right) , \end{aligned}$$
(4)

where \(c_s=1\), \(c_{t}=2+c_{t+1}+d_{t+1}\), and the sum of all entries is equal to n.

As before, for every matrix the sum of the entries of each column gives the respective part of the original partition. That is, the \(k^{th}\) part is equal to \(c_k+d_k\).

Now we apply the Path Procedure to the matrices from Theorem 5.1. As an example, we take \(n=6\) and show its partitions into 2-distinct parts with their associated matrices and corresponding partition into distinct odd parts.

Example 5.2

The Path Procedure applied to the partitions of 6 into 2-distinct parts generates the partitions into distinct odd parts given in Table 4.

Table 4 Table for Example 5.2

Remark 5.3

Note that now every integer partition generated by the Path Procedure has a part of size 1. This is easy to see since \(c_s=1\), which means that the first move in the path from \(P=(\sum _{i=1}^{s} d_i,\sum _{i=1}^{s} c_i)\) to (0, 0) is an exact down shift of size one.

Figure 7 below illustrates the distribution of frequencies of partitions of m in squares of size \(20\times 20\), induced by the partitions of \(n=20\), according to Theorem 5.1. Each cell contains how many partitions of m (indicated in the right down side of the cell) are generated by the matrix representation of the partitions of 20.

Fig. 7
figure7

\(n\times n\) square for \(n=20\)

In a similar way as done in Definition 4.1, the Path Procedure applied to the matrices of Theorem 5.1 motivates the following definition.

Definition 5.4

We call \(P_{R(od)}(m)\) the set of partitions of m into distinct odd parts, always having 1 as their smallest part, and whose size of any subsequence of consecutive odd integers equals the size of the previous subsequence of consecutive odd integers that were omitted before the subsequence started plus the size of the previous subsequence of consecutive odd parts plus 2. Also, \(\big \vert P_{R(od)}(m)\big \vert = p_{R(od)}(m)\).

Remark 5.5

If the part 3 appears after the part 1, this means that no odd integer was omitted. In this case, the next subsequence of odd parts will have size \(1+2=3\). If again after the first 4 odd parts the next one is exactly the fifth odd integer, then the next subsequence of odd parts will have size \(3+2=5\). In general, if no odd integer is omitted after some subsequence of parts, we assume the number of omitted parts is zero, and the size of the following subsequence of odd parts will be the size of the previous subsequence of consecutive odd parts plus 2.

By observing the partitions into distinct odd parts built by the Path Procedure we get the following results.

Proposition 5.6

The hooks induced by the order \(2\times 2\) matrices of Theorem 5.1 associated to the partitions of some n constitute partitions of \(3k^2+14k+16\), with \(0\le k\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -2\).

Example 5.7

For \(n=11\) there are 4 partitions into 2-distinct parts whose associated matrix has order \(2\times 2\), as shown in Table 5.

Table 5 Table for Example 5.7

Proposition 5.8

The hooks induced by the order \(2\times 3\) matrices of Theorem 5.1 with \(d_2=1\) or 0, associated to the partitions of some n, constitute partitions of \(m=\dfrac{4k^2+44k+117+3(-1)^k}{2}\), with \(1\le k\le \bigg \lfloor \dfrac{2n}{3}\bigg \rfloor -5\). Moreover, these partitions appear for the first time when \(n=\bigg \lfloor \dfrac{3k}{2}\bigg \rfloor +8\).

Example 5.9

For \(n=14\) there are 4 partitions into 2-distinct parts whose associated matrix has order \(2\times 3\) and entry \(d_2=1\) or 0 (Table 6).

Table 6 Table for Example 5.9

Proposition 5.8 can be generalized in such a way that an unique result characterizes all the matrices of order \(2\times 3\), not only those with entry \(d_2=0\) or 1. This is what the following theorem describes.

Theorem 5.10

The matrices of order \(2\times 3\) from Theorem 5.1, associated to the partitions of some n, have the form

$$\begin{aligned} \left( \begin{array}{ccc} k+t+5 &{} t+3 &{} 1 \\ n-9-2k-3t &{} k &{} t \end{array} \right) , \end{aligned}$$
(5)

with \(0\le t\le \bigg \lfloor \dfrac{n-9}{3} \bigg \rfloor \) and \(0\le k\le \bigg \lfloor \dfrac{n-9-3t}{2} \bigg \rfloor \).

Moreover, these matrices induce partitions into distinct odd parts of type

$$\begin{aligned} \mu= & {} (2(3t+2k+9)-1, \ldots , 2(2t+k+6)-1, 2(2t+k+5)-1, \nonumber \\&2(2t+4)-1, \ldots ,2(t+3)-1, 2(t+2)-1, 1). \end{aligned}$$
(6)

Proof

Let \(\lambda =(\lambda _1, \lambda _2, \lambda _3)\) be a partition of n into 2-distinct parts. So, \(\lambda _1\ge \lambda _2+2\) and \(\lambda _2\ge \lambda _3+2\). Let us call \(\lambda _2=\lambda _3+2+k\) and \(\lambda _3= 1+t\), with \(k,t\ge 0\). Then

$$\begin{aligned} \lambda _2=\lambda _3+2+k=k+t+3 \end{aligned}$$

and

$$\begin{aligned} \lambda _1=n-\lambda _2-\lambda _3=n-(k+t+3)-(1+t)=n-k-2t-4. \end{aligned}$$

It is easy to see that \(\lambda \) can be written as a matrix of type (5). And from the Path Procedure we clearly get the partition \(\mu \) given by (6). \(\square \)

In a similar way as done in the previous section, now we characterize a more general type of partitions into distinct odd parts generated by the Path Procedure applied to the partitions of n into 2-distinct parts.

Recall that the smallest odd part of any of these partitions is always 1, since the matrix representation of every partition of n into 2-distinct parts has entry \(c_s=1\). This means that the path from the line \(x+y=n\) to the origin starts with a down move of size 1.

In this case, the first subsequence of consecutive odd parts is composed exactly of one part of size 1. After it, we have the first missing subsequence of the partition, determined by the size of \(d_s\), which we call \(k_1-1\). The first missing subsequence of odd integers is

$$\begin{aligned} 2k_1-1, \ldots ,5,3. \end{aligned}$$

After the first missing subsequence of \(k_1-1\) consecutive odd integers we have the second subsequence of consecutive odd parts that compose the partition. Its size is determined by the entry \(c_{s-1}\) of the matrix. As \(c_{s-1}=2+c_s+d_s=2+1+k_1-1=k_1+2\), this means that the parts of the second subsequence are

$$\begin{aligned} 2(2k_1+2)-1,\ldots ,2(k_1+2)-1,2(k_1+1)-1. \end{aligned}$$

Some examples of partitions that have a part 1 followed by the first missing subsequence and after it exactly one subsequence of consecutive odd parts are

$$\begin{aligned} (7, 5, 3, 1), (15, 13, 11, 9, 7, 1), (23, 21, 19, 17, 15, 13, 11, 1), \end{aligned}$$

and

$$\begin{aligned} (35, 33, 31, 29, 27, 25, 23, 21, 19, 17, 1), \end{aligned}$$

which are, respectively, partitions of 16, 56, 120, and 261.

The following result gives a general characterization of which numbers are partitioned into distinct odd parts, its partition having one part of size 1 followed by the first missing subsequence and after it exactly one subsequence of consecutive odd parts.

Proposition 5.11

The Path Procedure applied to the partitions of n into 2-distinct parts generates partitions of \(m=3k_1^2+8k_1+5\), with \(1\le k_1\le \bigg \lfloor \dfrac{n-2}{2}\bigg \rfloor \), those being precisely all of the numbers whose partition has two subsequences of consecutive odd parts.

Proof

As mentioned before, the first subsequence of consecutive parts is always composed by a unique part of size 1. After it, the first missing subsequence, which might exist or not, has size \(d_s=k_1-1\). And the parts of the second subsequence are \(2(2k_1+2)-1,\ldots ,2(k_1+2)-1,2(k_1+1)-1\). So we may write m as

$$\begin{aligned} m= & {} 2(2k_1+2)-1+\cdots +2(k_1+2)-1+2(k_1+1)-1+1\\= & {} \frac{(2(2k_1+2)-1+2(k_1+1)-1)(k_1+2)}{2}+1 \\= & {} (3k_1+2)(k_1+2)+1 \\= & {} 3k_1^2+8k_1+5. \end{aligned}$$

So we get \(\mu =(2(2k_1+2)-1,\ldots ,2(k_1+2)-1,2(k_1+1)-1,1)\) a partition of \(m=3k_1^2+8k_1+5\). Clearly \(k_1\) has to be at most \(\dfrac{n-2}{2}\), otherwise the greatest part \(2(2k_1+2)-1\) would exceed \(2n-1\). And \(k_1\) has to be greater than or equal to 1 so we can have \(d_1=k_1-1=0\). \(\square \)

As it happens with unrestricted partitions, partitions of n into 2-distinct parts with more than two parts have a matrix representation into more than two columns, which means that each one of its \(c_t\) generates a different subsequence of consecutive odd parts.

We call the second missing subsequence the sequence of \(d_{s-1}=k_2\ge 0\) consecutive odd integers that do not appear as parts of the partition, which are

$$\begin{aligned}2(2k_1+k_2+2)-1,\ldots ,2(2k_1+4)-1, 2(2k_1+3)-1. \end{aligned}$$

Again, \(k_2\) can actually be equal to 0. Its size is determined by \(\lambda _{s-1}=c_{s-1}+d_{s-1}\), where \(\lambda _{s-1}\) is part of a partition of n into 2-distinct parts, and if \(\lambda _{s-1}=\lambda _s+2\) this means \(d_{s-1}=0\).

After the second missing subsequence of \(k_2\) consecutive odd integers we have the third subsequence of consecutive odd parts that compose the partition, determined by the size of entry \(c_{s-2}\) of the matrix. As \(c_{s-2}=2+c_{s-1}+d_{s-1}=2+2+k_1+k_2=k_1+k_2+4\), the parts of the second subsequence are

$$\begin{aligned} 2(3k_1+2k_2+6)-1,\ldots ,2(2k_1+k_2+2+2)-1, 2(2k_1+k_2+2+1)-1. \end{aligned}$$

For example,

$$\begin{aligned} (29, 27, 25, 23, 21, 19, 17, 15, 7, 5, 3, 1) \end{aligned}$$

and

$$\begin{aligned} (33, 31, 29, 27, 25, 23, 21, 19, 15, 13, 11, 9, 7, 1), \end{aligned}$$

which are, respectively, partitions of 192 and 264.

A general characterization of which numbers are partitioned into distinct odd parts, its partition having one part of size 1, the first and second missing subsequences and after each one of them a subsequence of consecutive odd parts, is given next.

Proposition 5.12

The Path Procedure applied to partitions of n into 2-distinct parts, having exactly three parts generates partitions of \(m=8k_1^2+3k_2^2+8k_1k_2+36k_1+20k_2+37\), with \(1\le k_1\le \bigg \lfloor \dfrac{n-2}{2}\bigg \rfloor \) and \(0\le k_2\le \bigg \lfloor \dfrac{n-3k_1-6}{2}\bigg \rfloor \).

Now let us consider a partition into distinct odd parts having tmissing subsequences and \(t+1\)subsequences of consecutive odd parts. We call \(k_1-1,k_2, \ldots ,k_t\) the sizes of the missing subsequences and, consequently, the subsequence after a missing subsequence of size \(k_i\) has size \(k_1+k_2+\cdots +k_i+2i\). The following lemma establishes the limits for each \(k_i\).

Lemma 5.13

Theith missing subsequence of a partition into distinct odd parts, whose parts derive from the Path Procedure applied to partitions of n into 2-distinct parts, is at most

$$\begin{aligned} \dfrac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1}+i(i+1))}{2}. \end{aligned}$$

Lemma 5.13 allows us to extend our construction to a more general characterization of the numbers partitioned into distinct odd parts, whose parts derive from the Path Procedure applied to partitions of n into 2-distinct parts.

Theorem 5.14

The partitions into distinct odd parts induced by the Path Procedure applied to partitions of n into 2-distinct parts are all of the form

$$\begin{aligned}&\sum _{\begin{array}{c} i=1 \\ 1\le k_1\le \frac{n-2}{2} \\ i>1,\, 0\le k_i\le \frac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1}+i(i+1))}{2} \end{array}}^{t}[(t+2-i)^2-1]k_i^2 \nonumber \\&\quad +\,\sum _{\begin{array}{c} i=1 \\ 1\le k_1\le \frac{n-2}{2} \\ i>1,\, 0\le k_i\le \frac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1}+i(i+1))}{2} \end{array}}^{t-1}k_{i}\sum _{j=1}^{t-i}j(2(t-i+3))k_{t-j+1} \nonumber \\&\quad +\, \sum _{i=1}^{t}k_i\big [2(t-i+1)t^2+6(t-i+1)t-2(t-i)(t-i+1)\big ]\nonumber \\&\quad +\, [t(t+1)]^2+1, \end{aligned}$$
(7)

where \(1\le t\le n-1\).

The Path Procedure Applied to Partitions Counted by the \(2^{nd}\) Rogers–Ramanujan Identity

In a similar way as done in Sect. 5, also in Santos et al. (2011) there is a result which characterizes a matrix representation for partitions into 2-distinct parts, greater than 1. These partitions are counted by the right-hand side of the \(2^{nd}\) Rogers–Ramanujan Identity,

$$\begin{aligned} p(n \vert \text{ parts } \equiv 2 \text{ or } 3\pmod 5)=p(n\vert 2\text{-distinct } \text{ parts }>1). \end{aligned}$$

Theorem 6.1

(Corollary 3.4, Santos et al. 2011) The number of partitions of n where the difference between parts is at least two and each part is greater than one is equal to the number of two-line matrices of the form

$$\begin{aligned} \left( \begin{array}{ccccc} c_1 &{} c_2 &{} c_3 &{} \cdots &{} c_s \\ d_1 &{} d_2 &{} d_3 &{} \cdots &{} d_s \end{array} \right) , \end{aligned}$$
(8)

where \(c_s=2\), \(c_{t}=2+c_{t+1}+d_{t+1}\), and the sum of all entries is equal to n.

The same Path Procedure is now applied to the matrices from Theorem 6.1, generating new integer partitions into distinct odd parts, as for example the ones shown below.

Example 6.2

The partitions of 8 into 2-distinct parts greater than 1 generate the following partitions into distinct odd parts (Table 7).

Table 7 Table for Example 6.2

Remark 6.3

Note that every integer partition generated by the hooks has a part of size 1 and a part of size 3. This is easy to see since \(c_s=2\), which means that the first move in the path from \(P=(\sum _{i=1}^{s} d_i,\sum _{i=1}^{s} c_i)\) to (0, 0) is an exact down shift of size two, generating the first two odd integers as parts of the new partition.

Again motivated by the Path Procedure applied to the matrices of Theorem 6.1 we have the following definition.

Definition 6.4

We call \(P_{R_2(od)}(m)\) the set of partitions of m into distinct odd parts, always having 1 and 3 as their smallest parts, and whose size of any subsequence of consecutive odd integers equals the size of the previous subsequence of consecutive odd integers that were omitted before the subsequence started plus the size of the previous subsequence of consecutive odd parts plus 2. Also, \(\big \vert P_{R_2(od)}(m)\big \vert =p_{R_2(od)}(m)\).

Remark 6.5

If no odd integer is omitted after some subsequence of parts, we assume the number of omitted parts is zero, and the size of the following subsequence of odd parts will be the size of the previous subsequence of consecutive odd parts plus 2.

The results obtained for these partitions are similar to the ones in previous sections.

Proposition 6.6

The hooks induced by the order \(2\times 2\) matrices of Theorem 6.1 associated to the partitions of some n constitute partitions of \(3k^2+20k+36\), with \(0\le k\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -3\).

Example 6.7

For \(n=13\) there are 4 partitions into 2-distinct parts greater than 1 whose associated matrix has order 2 (Table 8).

Table 8 Table for Example 6.7

Proposition 6.8

The Path Procedure applied to the order \(2\times 3\) matrices of Theorem 6.1 with \(d_2=0\), associated to the partitions of some n, generates partitions of \(m=8k^2+68k+144\), with \(0\le k\le \bigg \lfloor \dfrac{n}{3}\bigg \rfloor -4\). Moreover, these partitions appear for the first time when \(n=3k+12\).

Example 6.9

For \(n=25\) there are 5 partitions into 2-distinct parts greater than 1 whose associated matrix has order \(2\times 3\) and entry \(d_2=0\), as shown in Table 9.

Table 9 Table for Example 6.9

A similar result describes the order \(2\times 3\) matrices with \(d_2=1\).

Proposition 6.10

The Path Procedure applied to the order \(2\times 3\) matrices of Theorem 6.1 with \(d_2=1\), associated to the partitions of some n, generates partitions of \(m=8k^2+76k+183\), with \(0\le k\le \bigg \lfloor \dfrac{n-2}{3}\bigg \rfloor -4\). Moreover, these partitions appear for the first time when \(n=3k+14\).

Example 6.11

For \(n=25\) there are 4 partitions into 2-distinct parts greater than 1 whose associated matrix has order \(2\times 3\) and entry \(d_2=1\), as shown in Table 10.

Table 10 Table for Example 6.11

The Path Procedure applied to the partitions of n into 2-distinct parts greater than 1 is very similar to the one applied to partitions of n into 2-distinct parts of any size. In the present case the smallest parts of any partition into distinct odd parts are always 1 and 3, since the matrix representation of every partition of n into 2-distinct parts greater than 1 has entry \(c_s=2\). This means that the path from the line \(x+y=n\) to the origin starts with a down move of size 2.

In this case, the first subsequence of consecutive odd parts is composed of one part of size 1 and one part of size 3. After it, we have the first missing subsequence of the partition, determined by the size of \(d_s\), which we call \(k_1-2\). The first missing subsequence of parts is

$$\begin{aligned} 2k_1-1, \ldots ,7,5. \end{aligned}$$

After the first missing subsequence of \(k_1-2\) consecutive odd integers we have the second subsequence of consecutive odd parts that compose the partition. Its size is determined by the entry \(c_{s-1}\) of the matrix. As \(c_{s-1}=2+c_s+d_s=2+2+k_1-2=k_1+2\), this means that the parts of the second subsequence are

$$\begin{aligned} 2(2k_1+2)-1,\ldots ,2(k_1+2)-1,2(k_1+1)-1. \end{aligned}$$

Some examples of partitions that have a part 1 and a part 3 followed by the first missing subsequence and after it exactly one subsequence of consecutive odd parts are

$$\begin{aligned} (11, 9, 7, 5, 3, 1), (19, 17, 15, 13, 11, 9, 3, 1), \text{ and } (27, 25, 23, 21, 19, 17, 15, 13, 3, 1), \end{aligned}$$

partitions of 36, 88, and 164, respectively.

The following result gives a general characterization of which numbers are partitioned into distinct odd parts, its partition having one part of size 1 and one part of size 3, followed by the first missing subsequence and after it exactly one subsequence of consecutive odd parts.

Proposition 6.12

The Path Procedure applied to the partitions of n into 2-distinct parts greater than 1 generates partitions of \(m=3k_1^2+8k_1+8\), with \(2\le k_1\le \bigg \lfloor \dfrac{n-2}{2}\bigg \rfloor \), those being precisely all of the numbers whose partition has two subsequences of consecutive odd parts.

As it happens in the cases explored in previous sections, partitions of n into 2-distinct parts greater than 1 with more than two parts have a matrix representation into more than two columns, which means that each one of its \(c_t\) generates a different sequence of consecutive odd parts.

We call the second missing subsequence the sequence of \(d_{s-1}=k_2\ge 0\) consecutive odd integers that do not appear as parts of the partition, which are

$$\begin{aligned} 2(2k_1+k_2+2)-1,\ldots ,2(2k_1+4)-1, 2(2k_1+3)-1. \end{aligned}$$

Again, \(k_2\) can actually be equal to 0. Its size is determined by \(\lambda _{s-1}=c_{s-1}+d_{s-1}\), where \(\lambda _{s-1}\) is part of a partition of n into 2-distinct parts greater than 1, and if \(\lambda _{s-1}=\lambda _s+2\) this means \(d_{s-1}=0\).

After the second missing subsequence of \(k_2\) consecutive odd integers we have the third subsequence of consecutive odd parts that compose the partition, determined by the size of entry \(c_{s-2}\) of the matrix. As \(c_{s-2}=2+c_{s-1}+d_{s-1}=2+k_1+2+k_2=k_1+k_2+4\), the parts of the second subsequence are

$$\begin{aligned} 2(3k_1+2k_2+6)-1,\ldots ,2(2k_1+k_2+2+2)-1, 2(2k_1+k_2+2+1)-1. \end{aligned}$$

For example,

$$\begin{aligned} (31, 29, 27, 25, 23, 21, 19, 17, 11, 9, 7, 5, 3, 1) \end{aligned}$$

and

$$\begin{aligned} (33, 31, 29, 27, 25, 23, 21, 19, 15, 13, 11, 9, 7, 3, 1), \end{aligned}$$

which are, respectively, partitions of 228 and 267.

A general characterization of which numbers are partitioned into distinct odd parts, its partition having one part of size 1, one part of size 3, the first and second missing subsequences and after each one of them a subsequence of consecutive odd parts, is given next.

Proposition 6.13

The Path Procedure applied to the partitions of n into 2-distinct parts, greater than 1, having exactly three parts generates partitions of \(m=8k_1^2+3k_2^2+8k_1k_2+36k_1+20k_2+40\), with \(2\le k_1\le \bigg \lfloor \dfrac{n-2}{2}\bigg \rfloor \) and \(0\le k_2\le \bigg \lfloor \dfrac{n-3k_1-6}{2}\bigg \rfloor \).

Again let us consider a partition into distinct odd parts having tmissing subsequences of sizes \(k_1-2,k_2, \ldots ,k_t\), and \(t+1\)subsequences of consecutive odd parts of sizes \(k_1+k_2+\cdots +k_i+2i\). The following lemma establishes the limits for each \(k_i\).

Lemma 6.14

Theith missing subsequence of a partition into distinct odd parts, whose parts derive from the Path Procedure applied to partitions of n into 2-distinct parts greater than 1, is at most

$$\begin{aligned} \dfrac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1}+i(i+1))}{2}. \end{aligned}$$

With the Lemma we can extend our construction to a more general characterization of the numbers partitioned into distinct odd parts, whose parts derive from the Path Procedure applied to partitions of n into 2-distinct parts greater than 1.

Theorem 6.15

The partitions into distinct odd parts induced by the Path Procedure applied to partitions of n into 2-distinct parts greater than 1, are all of the form

$$\begin{aligned}&\sum _{\begin{array}{c} i=1 \\ 2\le k_1\le \frac{n-2}{2} \\ i>1,\, 0\le k_i\le \frac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1}+i(i+1))}{2} \end{array}}^{t}[(t+2-i)^2-1]k_i^2 \nonumber \\&\qquad +\,\sum _{\begin{array}{c} i=1 \\ 2\le k_1\le \frac{n-2}{2} \\ i>1,\, 0\le k_i\le \frac{n-((i+1)k_1+ik_2+\cdots +4k_{i-2}+3k_{i-1}+i(i+1))}{2} \end{array}}^{t-1}k_{i}\sum _{j=1}^{t-i}j(2(t-i+3))k_{t-j+1} \nonumber \\&\qquad +\, \sum _{i=1}^{t}k_i\big [2(t-i+1)t^2+6(t-i+1)t-2(t-i)(t-i+1)\big ]\nonumber \\&\qquad +\, [t(t+1)]^2+4, \end{aligned}$$
(9)

where \(1\le t\le n-1\).

Remark 6.16

Observe that the matrix representation for partitions of n into 2-distinct parts greater than 1 is exactly the same as the one for partitions generated by the unsigned version of mock theta function

$$\begin{aligned} f_1(q)=\sum _{n=0}^{\infty }\frac{q^{n^2+n}}{(-q;q)_n}, \end{aligned}$$

according to the table in page 240 of Brietzke et al. (2013).

The general term

$$\begin{aligned} \frac{q^{2(1+2+\cdots +s)}}{(1-q)(1-q^2)\cdots (1-q^s)} \end{aligned}$$

of the unsigned version \(f_{1^*}(q)\) of function \(f_1(q)\) counts the partitions of n containing at least 2 copies of each part from 1 to s. By conjugation, this is the same as counting the partitions of n into 2-distinct parts greater than 1. Further informations about function \(f_{1^*}(q)\) can be found in Bagatini et al. (2017).

The Path Procedure Applied to Partitions Counted by the Mock Theta Function \(T_1(-q)\)

In this section we rapidly describe which partitions into distinct odd parts are obtained from the Path Procedure applied to the partitions generated by mock theta function \(T_1(-q)\), that is, the unsigned version of \(T_1(q)\) (see Brietzke et al. 2013).

So, let us consider the mock theta function

$$\begin{aligned} T_1(q)=\sum _{n=0}^{\infty }\frac{q^{n(n+1)}(-q^2,q^2)_n}{(-q,q^2)_{n+1}}. \end{aligned}$$

Its unsigned version

$$\begin{aligned} T_1^*(q):=T_1(-q)=\sum _{n=0}^{\infty }\frac{q^{n(n+1)}(-q^2,q^2)_n}{(q,q^2)_{n+1}} \end{aligned}$$

has general term

$$\begin{aligned} \frac{q^{2+4+\cdots +2s}(1+q^2)(1+q^4)\cdots (1+q^{2s})}{(1-q)(1-q^3)\cdots (1-q^{2s+1})}, \end{aligned}$$

which counts the partitions of n containing one or two parts equal to each one of the even numbers from 2 to 2s, and any number of odd parts less than or equal to \(2s+1\).

Example 7.1

The partitions of 10 counted by \(T_1^*(q)\) are

$$\begin{aligned}&(1, 1, 1, 1, 1, 1, 1, 1, 1, 1),(2,1, 1, 1, 1, 1, 1, 1, 1),(2,2,1, 1, 1, 1, 1, 1), \\&\quad (3,2,1, 1, 1, 1, 1),(3,2,2,1, 1, 1)(3,3,2,1, 1),(3,3,2, 2), \\&\quad (4,2,1, 1, 1, 1),(4,2,2,1, 1),(4,3,2,1)\text{, } \text{ and } (4,4,2). \end{aligned}$$

According to Brietzke et al. (2013), the partitions generated by \(T_1^*(q)\) can also be expressed in terms of two-line matrices. We describe this matrix representation in the following theorem.

Theorem 7.2

(Table p. 243, Brietzke et al. 2013) The number of partitions of n generated by \(T_1^*(q)\) is equal to the number of two-line matrices of the form

$$\begin{aligned} \left( \begin{array}{ccccc} c_1 &{} c_2 &{} \cdots &{} c_s &{} c_{s+1} \\ d_1 &{} d_2 &{} \cdots &{} d_s &{} d_{s+1} \end{array} \right) , \end{aligned}$$
(10)

where \(c_{s+1}=0\), \(d_t\ge 0\), \(c_{t}=j_t+c_{t+1}+2d_{t+1}\), with \(j_t\in \{2,4\}\), and the sum of all entries is equal to n.

Proof

If we write

$$\begin{aligned} n= & {} 2\cdot (1+j_1)+4\cdot (1+j_2)+\cdots +2s\cdot (1+j_s)+1\cdot d_1+3\cdot d_2\\&+\cdots +(2s+1)\cdot d_{s+1}, \end{aligned}$$

with \(j_t\in \{0,1\}\) and \(d_t\ge 0\), we can easily organize this sum in a two-line matrix like

$$\begin{aligned} \left( \begin{array}{cccc} \mathop {\sum }\nolimits _{t=1}^s2(1+j_t)+\mathop {\sum }\nolimits _{t=2}^{s+1}2d_t &{} \cdots &{} 2(1+j_s)+2d_{s+1} &{} 0 \\ d_1 &{} \cdots &{} d_s &{} d_{s+1} \end{array} \right) , \end{aligned}$$
(11)

whose entries satisfy exactly the conditions we needed. \(\square \)

Remark 7.3

Note that the number of columns in the matrix representation (10) equals the number of different even parts plus one. Differently from the representations in previous sections, where the number of columns was the same as the number of parts of the partitions, in the present case we can say that the first line of the matrix counts the even parts and the second line counts the odd parts. At least at first the values \(c_t+d_t\) have no interesting significance to us.

Now we apply the Path Procedure to the set of matrices from Theorem 7.2 and create partitions of integers m into distinct odd parts.

Example 7.4

The partitions of 10 counted by function \(T_1^*(q)\) have the following matrix representations, which generate the partitions into distinct odd parts, given in Table 11.

Table 11 Table for Example 7.4

Remark 7.5

In every matrix like (10) we have \(c_{s+1}=0\), which means that the entry which determines the first move in the path from \(x+y=n\) to (0, 0) is \(d_{s+1}\). Although, as \(d_{s+1}\) may be zero, in this case the first move is determined by the entry \(c_s\). So, if \(d_{s+1}>0\), the partition has \(2d_{s+1}+1\) as its smallest part, and if \(d_{s+1}=0\), the smallest part of the partition is 1.

When applied to the matrices of Theorem 7.2, the Path Procedure motivates the next definition.

Definition 7.6

We call \(P_{T(od)}(m)\) the set of partitions of m into distinct odd parts greater than or equal to 1 whose size of any subsequence of consecutive odd integers equals two times the size of the previous subsequence of consecutive odd integers that were omitted before the subsequence started plus the size of the previous subsequence of consecutive odd parts plus 2 or 4. Also, \(\big \vert P_{T(od)}(m)\big \vert =p_{T(od)}(m)\).

Remark 7.7

If no odd integer is omitted after some subsequence of parts, we assume the number of omitted parts is zero, and the size of the following subsequence of odd parts will be the size of the previous subsequence of consecutive odd parts plus 2 or 4.

In order to describe what kind of partitions into distinct odd parts may be obtained from the Path Procedure applied to partitions generated by \(T_1^*(q)\), first recall that, according to Remark 7.5, if a partition of n generated by the function \(T_1^*(q)\) has an even part as its greatest one, then its matrix representation has \(d_{s+1}=0\). Consequently, the corresponding partition into distinct odd parts via Path Procedure has 1 as its smallest part.

On the other hand, if the greatest part of a partition of n generated by the function \(T_1^*(q)\) is odd, then its matrix representation has \(d_{s+1}>0\), and the smallest part of the corresponding partition into distinct odd parts via Path Procedure is \(2d_{s+1}+1\) (see for example partitions (2, 2, 1, 1, 1, 1, 1, 1) and (3, 3, 2, 2) of 10 in Example 7.4).

As the smallest part of a generated partition into distinct odd parts may be any integer greater than or equal to 1, we have the first missing subsequence of \(k_1\ge 0\) consecutive odd integers. After it, the first subsequence of parts of the partition has size \(j_1+2k_1\), with \(j_1\in \{2,4\}\). That is, the sequence of parts \(2(k_1+1)-1,2(k_1+2)-1,\cdots 2(3k_1)-1, 2(3k_1+1)-1, 2(3k_1+2)-1\) and maybe \(2(3k_1+3)-1\) and \(2(3k_1+4)-1\).

It is not difficult to choose between \(j_1=2\) or \(j_1=4\) when, after the first subsequence of parts, the second missing subsequence has size \(k_2>0\). If \(k_2=0\), we need to verify the size of the second subsequence of parts, which is \(j_2+(j_1+2k_1)+2k_2\), with \(j_2\in \{2,4\}\).

This process goes on until the end of the sequence of different odd parts of the partition, paying attention to the fact that the last entry \(d_1\) depends on the number n, whose original partition induced the partition into distinct odd parts.

Some numbers that appear as partitions into distinct odd parts, induced by the Path Procedure applied to the function \(T_1^*(q)\), are described in the following result.

Proposition 7.8

The Path Procedure applied to partitions generated by the mock theta function \(T_1^*(q)\) induces partitions of \((2k)^2\) into distinct odd parts.

Proof

We may partition \((2k)^2\) as

$$\begin{aligned} (2(2k)-1,2(2k-1)-1,\ldots ,3,1), \end{aligned}$$

that is, in 2k consecutive odd parts. This means that all the missing subsequences have size zero, and the second line of the matrix which originated the partition of \((2k)^2\) has all its entries equal to zero (except possibly for \(d_1\), which does not generate any part). So, the partition of \((2k)^2\) is determined by the first line of the matrix, which has even decreasing entries from \(c_1\) to \(c_{s+1}\) whose difference between consecutive entries equals 2 or 4.

If \(2k=t(t+1)\) for some \(t\in {\mathbb {N}}\), we can write

$$\begin{aligned} 2k=t(t+1)=2t+(2t-2)+\cdots +4+2, \end{aligned}$$

and we get 2t, \(2t-2\), \(\ldots \), 4, 2, and 0 as the entries \(c_1\), \(c_2\), \(\ldots \), \(c_s\), and \(c_{s+1}\).

If \(2k=t(t+1)+2l\) for \(1\le l\le t\), we take the values of \(c_1\), \(c_2\), \(\ldots \), \(c_s\), and \(c_{s+1}\) from the previous case and add 2 to the first l entries \(c_i\).

So we may write

$$\begin{aligned} 2k= & {} t(t+1)+2l=2(t+1)+(2(t+1)-2)+\cdots +(2(t+1)-2(l-1)) \\&+\,(2(t+1)-2(l+1)) +\cdots +4+2, \end{aligned}$$

and we get \(2(t+1)\), \(2(t+1)-2\), \(\ldots \), \(2(t+1)-2(l-1)\),\(2(t+1)-2(l+1)\), \(\ldots \), 4, 2, and 0 as the entries \(c_1\), \(c_2\), \(\ldots \), \(c_s\), and \(c_{s+1}\). \(\square \)

Example 7.9

Take the partitions (2, 1, 1, 1, 1, 1, 1, 1, 1), (2, 2, 1, 1, 1, 1, 1, 1), \((4,2,1,1,1,1)\), (4, 2, 2, 1, 1), and (4, 4, 2) of 10 from Example 7.4. They generate partitions of \(2^2\), \(4^2\), \(6^2\), \(8^2\), and \(10^2\), respectively.

The Path Procedure Applied to Different Matrix Representations for Unrestricted Partitions

Besides the representation given by Theorem 2.1, unrestricted integer partitions have at least two other matrix representations, given in Santos et al. (2011).

Theorem 8.1

(Theorem 4.3, Santos et al. 2011). The number of unrestricted partitions of n is equal to the number of two-line matrices of the form

$$\begin{aligned} \left( \begin{array}{ccccc} c_1 &{} c_2 &{} c_3 &{} \cdots &{} c_s \\ d_1 &{} d_2 &{} d_3 &{} \cdots &{} d_s \end{array} \right) , \end{aligned}$$
(12)

where \(d_t\ne 0\), \(c_{t}\ge 1+c_{t+1}+d_{t+1}\), and the sum of all entries is equal to n.

Theorem 8.2

(Corollary 4.5, Santos et al. 2011) The number of unrestricted partitions of n is equal to the number of two-line matrices of the form

$$\begin{aligned} \left( \begin{array}{ccccc} c_1 &{} c_2 &{} c_3 &{} \cdots &{} c_s \\ d_1 &{} d_2 &{} d_3 &{} \cdots &{} d_s \end{array} \right) , \end{aligned}$$
(13)

where \(c_s\ne 0\), \(c_{t}\ge 2+c_{t+1}+d_{t+1}\), and the sum of all entries is equal to n.

The bijective proofs of both theorems can be found in Brietzke et al. (2010). Differently from the first matrix representation, studied in Sect. 4, the number s of columns in matrices 12 and 13 equals the size of the side of the Durfee square of the associated partition. As an example, we take \(n=5\) and show all of its partitions with their associated matrices of types 12 and 13.

Example 8.3

For \(n=5\) we have \(p(5)=7\), and so there are 7 matrices satisfying Theorems 8.1 and 8.2, as shown in Table 12.

Table 12 Table for Example 8.3

We now apply the Path Procedure to the matrices associated to the partitions of n, generating partitions of \(m\le n^2\) into distinct odd parts.

Example 8.4

The matrices associated to the partitions of \(n=5\) from Example 8.3 generate the partitions into distinct odd parts contained in Table 13.

Table 13 Table for Example 8.4

The results obtained form the observation of the partitions generated by the Path Procedure applied to the matrices from Theorems 8.1 and 8.2 are the following.

Theorem 8.5

The Path Procedure applied to the order \(2\times 2\) matrices of Theorem 8.1 generates partitions of every m of the form \(m=j^2-k^2\), for \(0\le k\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\) and \(2k+1\le j\le n-1\).

Proof

Note that

$$\begin{aligned} j^2-k^2= & {} (j+k)(j-k)\\= & {} \dfrac{(2(k+1)-1+2j-1)(j-(k+1)+1)}{2}\\= & {} (2(k+1)-1)+(2(k+2)-1)+\cdots +(2(j-1)-1)+(2j-1). \end{aligned}$$

If seen as a partition, the sequence \(((2j-1),(2(j-1)-1),\ldots ,(2(k+2)-1),(2(k+1)-1))\) is generated by the Path Procedure applied to a matrix of order \(2\times 2\), having entries \(c_2=0\), \(d_2=k\), \(c_1=j-(k+1)+1=j=k\), and \(d_1=n-c_2-d_2-c_1=n-j\). Observe that

$$\begin{aligned} c_1\ge 1+c_2+d_2\Longrightarrow j-k\ge 1+k\Longrightarrow j\ge 2k+1 \end{aligned}$$

and

$$\begin{aligned} d_1\ne 0\Longrightarrow n-j\ge 1\Longrightarrow n-1\ge j. \end{aligned}$$

So we get the limitation \(2k+1\le j\le n-1\).

We get the limitations for k by observing that, if we had \(k>\bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\), then

$$\begin{aligned} k>\bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\Longrightarrow k\ge \bigg \lfloor \dfrac{n}{2}\bigg \rfloor \Longrightarrow j\ge 2\bigg \lfloor \dfrac{n}{2}\bigg \rfloor +1\ge n, \end{aligned}$$

which is a contradiction. \(\square \)

Theorem 8.6

The Path Procedure applied to the order \(2 \times 2\) matrices of Theorem 8.2 generates partitions of every m of the form \(m=1+j^2-k^2\), for \(1\le k\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\) and \(2k+2\le j\le n\).

Proof

Note that

$$\begin{aligned} 1+j^2-k^2= & {} 1+(j+k)(j-k)\\= & {} 1+\dfrac{(2(k+1)-1+2j-1)(j-(k+1)+1)}{2}\\= & {} 1+(2(k+1)-1)+(2(k+2)-1)+\cdots \\&+(2(j-1)-1)+(2j-1). \end{aligned}$$

If seen as a partition, the sequence \(((2j-1),(2(j-1)-1)\ldots ,(2(k+2)-1),(2(k+1)-1),1)\) is generated by the Path Procedure applied to a matrix of order \(2\times 2\), having entries \(c_2=1\), \(d_2=k-1\), \(c_1=j-(k+1)+1=j-k\), and \(d_1=n-c_2-d_2-c_1=n-j\). Observe that

$$\begin{aligned} c_1\ge 2+c_2+d_2\Longrightarrow j-k\ge 2+1+k-1\Longrightarrow j\ge 2k+2 \end{aligned}$$

and

$$\begin{aligned} d_1\ge 0\Longrightarrow n-j\ge 0\Longrightarrow n\ge j. \end{aligned}$$

So we get the limitation \(2k+2\le j\le n\).

Moreover,

$$\begin{aligned} d_2\ge 0\Longrightarrow k-1\ge 0\Longrightarrow k\ge 1 \end{aligned}$$

and if we had \(k>\bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\), then

$$\begin{aligned} k>\bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\Longrightarrow k\ge \bigg \lfloor \dfrac{n}{2}\bigg \rfloor \Longrightarrow j\ge 2\bigg \lfloor \dfrac{n}{2}\bigg \rfloor +1\ge n, \end{aligned}$$

which is a contradiction. So, \(1\le k\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\). \(\square \)

With analogous arguments, we can set a more general result, as it follows.

Theorem 8.7

The Path Procedure applied to the order \(2\times 2\) matrices of Theorem 8.1 generates precisely partitions of \(m=t^2+j^2-k^2\), for \(0\le t\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -2\), \(t+1\le k\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\) and \(2k+1\le j\le n-1\).

Proof

Note that

$$\begin{aligned} t^2+j^2-k^2= & {} t^2+(j+k)(j-k)\\= & {} 1+3+\cdots +2t-1+\dfrac{(2(k+1)-1+2j-1)(j-(k+1)+1)}{2}\\= & {} 1+3+\cdots +2t-1+(2(k+1)-1)+(2(k+2)-1) \\&+\,\cdots +(2j-1). \end{aligned}$$

If seen as a partition, the sequence \(((2j-1),\ldots ,(2(k+2)-1),(2(k+1)-1),(2t-1)\ldots ,3,1)\) is generated by the Path Procedure applied to a matrix of order \(2\times 2\), having entries \(c_2=t\), \(d_2=k-t\), \(c_1=j-k\), and \(d_1=n-j\). Clearly the limits set for t, k, and j satisfy the conditions we need, but why are these limitations precisely the exact ranges for t, k, and j?

Let us suppose \(t>\bigg \lfloor \dfrac{n}{2}\bigg \rfloor -2\), saying \(t=\bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1+r\), with \(r\ge 0\). As \(k\ge t+1\), then \(k\ge \bigg \lfloor \dfrac{n}{2}\bigg \rfloor +r\), which implies \(j\ge 2\bigg \lfloor \dfrac{n}{2}\bigg \rfloor +2r+1\ge n\). But then the entry \(d_1\) can only be 0, which contradicts the conditions of Theroem 8.1.

Moreover, as we are dealing with matrices of order \(2\times 2\), the original partition of n associated to each matrix has Durfee square of side 2. So, the generating function for these partitions of n is

$$\begin{aligned} \sum _{n=0}^{\infty }a(n)q^n= & {} \dfrac{q^4}{(1-q)^2(1-q^2)^2}\nonumber \\= & {} -\dfrac{1}{8(1+q)}+\dfrac{1}{16(1+q)^2}-\dfrac{1}{8(1-q)}+\dfrac{11}{16(1-q)^2}-\dfrac{3}{4(1-q)^3}\nonumber \\&+\,\dfrac{1}{4(1-q)^4}\nonumber \\= & {} \sum _{n=0}^{\infty }\bigg (-\dfrac{1}{8}(-1)^n+\dfrac{1}{16}(-1)^n(n+1)-\dfrac{1}{8}+\dfrac{11}{16}(n+1)-\dfrac{3}{4}\left( {\begin{array}{c}n+2\\ 2\end{array}}\right) \nonumber \\&+\,\dfrac{1}{4}\left( {\begin{array}{c}n+3\\ 3\end{array}}\right) \bigg )q^n \nonumber \\= & {} \sum _{n=0}^{\infty }\dfrac{1}{48}(n-1)(3(-1)^n-3-4n+2n^2)q^n. \end{aligned}$$
(14)

It turns out that the coefficient a(n) of \(q^n\) in expression (14) is exactly

$$\begin{aligned} \sum _{t=0}^{\big \lfloor \tfrac{n}{2}\big \rfloor -2}\sum _{k=t+1}^{\big \lfloor \tfrac{n}{2}\big \rfloor -1}\sum _{j=2k+1}^{n-1}1, \end{aligned}$$

which is \(\dfrac{(n+1)(n-1)(n-3)}{24}\) for odd n, and \(\dfrac{n(n-1)(n-2)}{24}\) for even n. \(\square \)

An analogous result is valid for the \(2\times 2\) matrices of Theorem 8.2.

Theorem 8.8

The Path Procedure applied to the order \(2\times 2\) matrices of Theorem 8.2 generates precisely partitions of \(m=t^2+j^2-k^2\), for \(1\le t\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\), \(t\le k\le \bigg \lfloor \dfrac{n}{2}\bigg \rfloor -1\) and \(2k+2\le j\le n\).

Proof

The result is true once we note that a bijection between the matrices of Theorems 8.1 and 8.2 makes

$$\begin{aligned} \left( \begin{array}{cc} j-k&{}t \\ n-j&{}k-t \end{array} \right) \longmapsto \left( \begin{array}{cc} j-k+1&{}t+1 \\ n-j-1&{}k-t-1 \end{array} \right) \end{aligned}$$

\(\square \)

Remark 8.9

Once a set of integer partitions has a matrix representation, it is possible to apply the Path Procedure to them. Therefore, the Path Procedure turns out being a promising road in the study of integer partitions and partition identities, as already the two-line matrix representation for different sets of partitions is. Clearly the results registered in this paper do not cover all of the possibilities. Many more results may be conjectured by observing the partitions into distinct odd parts generated by the Path Procedure applied to different sets of partitions.

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Correspondence to M. L. Matte.

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Santos, J.P.O., Matte, M.L. A New Approach to Integer Partitions. Bull Braz Math Soc, New Series 49, 811–847 (2018). https://doi.org/10.1007/s00574-018-0082-z

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Keywords

  • Integer partitions
  • Mock theta functions
  • Matrix representation
  • Partition identities

Mathematics Subject Classification

  • Primary 11P81
  • Secondary 05A19