## Abstract

Refined versions, analytic and combinatorial, are given for classical integer partition theorems. The examples include the Rogers–Ramanujan identities, the Göllnitz–Gordon identities, Euler’s odd = distinct theorem, and the Andrews–Gordon identities. Generalizations of each of these theorems are given where a single part is “marked” or weighted. This allows a single part to be replaced by a new larger part, “shifting” a part, and analogous combinatorial results are given in each case. Versions are also given for marking a sum of parts.

## Keywords

Partition Rogers–Ramanujan identities## Mathematics Subject Classification

Primary 05A17 Secondary 11P84## 1 Introduction

*w*, the factor \(1/(1-q^9)\) is replaced by

The result is known [9, (2.2)], and there is a refined combinatorial version. The key to the combinatorial result is that the terms in the sum side are positive as power series in *q* and *w*.

### Theorem 1.1

Here is a combinatorial version of Theorem 1.1.

### Theorem 1.2

*M*be a positive integer which is congruent to 1 or \(4\mod 5\). Then, the number of partitions of

*n*into parts congruent to 1 or \(4\mod 5\) with exactly

*k*

*M*’s is equal to the number of partitions \(\lambda \) of

*n*with difference at least 2 and

- 1.
if \(\lambda \) has one part, then \(\lfloor n/M\rfloor =k,\)

- 2.
if \(\lambda \) has at least two parts, then \(\lfloor (\lambda _1-\lambda _2-2)/M\rfloor =k.\)

The purpose of this paper is to give the analogous results for several other classical partition theorems: the Göllnitz–Gordon identities, Euler’s odd = distinct theorem, and the Andrews–Gordon identities. The main engine, Proposition 3.1, may be applied to many other single sum identities. The results obtained here by marking a part are refinements of the corresponding classical results.

Finally in Sect. 6, we consider marking a sum of parts. We can extend Theorem 1.2 to allow other values of *M*, for example \(M=7\), by marking the partition \(6+1\). See Corollary 6.8.

*q*is understood, we may write \((A;q)_k\) as \((A)_k.\)

## 2 The Rogers–Ramanujan Identities

In this section, we give prototypical examples for the Rogers–Ramanujan identities.

First, we state a marked version of the second Rogers–Ramanujan identity, which follows from Proposition 3.1.

### Theorem 2.1

Here is a combinatorial version of Theorem 2.1.

### Theorem 2.2

*M*be a positive integer which is congruent to 2 or \(3\mod 5\). Then, the number of partitions of

*n*into parts congruent to 2 or \(3\mod 5\) with exactly

*k*

*M*’s is equal to the number of partitions \(\lambda \) of

*n*with difference at least 2, no 1’s, and

- 1.
if \(\lambda \) has one part, then \(n=Mk+j\), \(2\le j\le M-1\), or \(j=0\) or \(j=M+1,\)

- 2.
if \(\lambda \) has at least two parts, then \(\lfloor (\lambda _1-\lambda _2-2)/M\rfloor =k.\)

### Proof

*k*-th term on the sum side

### Example 2.3

*M*is marked on the sum side. We need notation for when a double staircase is removed from a partition with difference at least two.

### Definition 2.4

For any partition \(\lambda \) with *k* parts whose difference of parts is at least 2, let \(\lambda ^*\) denote the partition obtained upon removing the double staircase \((2k-1,2k-3,\ldots ,1)\) from \(\lambda \), and reading the result by columns.

For any partition \(\lambda \) with *k* parts and no 1’s whose difference of parts is at least 2, let \(\lambda ^{**}\) denote the partition obtained upon removing the double staircase \((2k,2k-2,\ldots ,2)\) from \(\lambda \), and reading the result by columns.

### Theorem 2.5

*M*be a positive integer which is congruent to 1 or \(4\mod 5\). Then, the number of partitions of

*n*into parts congruent to 1 or \(4\mod 5\) with exactly

*k*

*M*’s is equal to the number of partitions \(\lambda \) of

*n*with difference at least 2 and

- 1.
if \(\lambda \) has one part, then \(\lfloor n/M\rfloor =k,\)

- 2.
if \(\lambda \) has between two and \(M-1\) parts, then \(\lfloor (\lambda _1-\lambda _2-2)/M\rfloor =k\),

- 3.
if \(\lambda \) has at least

*M*parts, then \(\lambda ^*\) has exactly*k**M*’s.

### Example 2.6

### Theorem 2.7

*M*be a positive integer which is congruent to 2 or \(3\mod 5\). Then, the number of partitions of

*n*into parts congruent to 2 or \(3\mod 5\) with exactly

*k*

*M*’s is equal to the number of partitions \(\lambda \) of

*n*with difference at least 2, no 1’s and

- 1.
if \(\lambda \) has one part, then \(n=Mk+j\), where \(2\le j\le M-1\), or \(j=0\) or \(j=M+1,\)

- 2.
if \(\lambda \) has between two and \(M-1\) parts, then \(\lfloor (\lambda _1-\lambda _2-2)/M\rfloor =k\),

- 3.
if \(\lambda \) has at least

*M*parts, then \(\lambda ^{**}\) has exactly*k**M*’s.

## 3 A General Expansion

In this section, we give a general expansion, Proposition 3.1, for marking a single part.

*q*.

- 1.
the Rogers–Ramanujan identities, \(\alpha _j=q^{j^2} {\text { or }} q^{j^2+j},\)

- 2.
Euler’s odd=distinct theorem, \(\alpha _j=q^{\left( {\begin{array}{c}j+1\\ 2\end{array}}\right) },\)

- 3.
the Göllnitz–Gordon identities,

*q*replaced by \(q^2\), \(\alpha _j=q^{j^2}(-q;q^2)_j,\) - 4.
all partitions by the largest part, \(\alpha _j=q^j,\)

- 5.
all partitions by Durfee square, \(\alpha _j=q^{j^2}/(q;q)_j.\)

*M*may be marked in general using the next proposition.

### Proposition 3.1

*M*, if \(\alpha _0=1\),

*q*, the right side has a combinatorial interpretation.

*M*is marked by

*w*.

*M*’s in \(\lambda ^*.\) Since

### 3.1 Distinct Parts

Choosing \(\alpha _j=q^{\left( {\begin{array}{c}j+1\\ 2\end{array}}\right) }\) in Proposition 3.1 gives distinct partitions, which by Euler’s theorem are equinumerous with partitions into odd parts. Here is the marked version.

### Corollary 3.2

*M*,

### Definition 3.3

For any partition \(\lambda \) with *j* distinct parts, let \(\lambda ^{St}\) be the partition obtained upon removing a staircase \((j,j-1,\ldots ,1)\) from \(\lambda \), and reading the result by columns.

### Example 3.4

If \(\lambda =(8,7,3,1)\), then \(\lambda ^{St}=(3,2,2,2).\)

Here is the combinatorial version of Corollary 3.2, generalizing Euler’s theorem.

### Theorem 3.5

*M*, the number of partitions of

*n*into odd parts with exactly

*k*parts of size

*M*is equal to the number of partitions \(\lambda \) of

*n*into distinct parts such that

- 1.
if \(\lambda \) has one part, then \(\lfloor n/M\rfloor =k,\)

- 2.
if \(\lambda \) has at least two parts, then \(\lfloor (\lambda _1-\lambda _2-1)/M\rfloor =k.\)

### Example 3.6

### Proposition 3.7

There is an *M*-version of the Sylvester “fishhook” bijection which proves Theorem 3.5.

### Proof

Let *FH* be the fishhook bijection from partitions with distinct parts to partitions with odd parts. If \(FH(\lambda )=\mu ,\) it is known that the number of 1’s in \(\mu \) is \(\lambda _1-\lambda _2-1\), except for \(FH(n)= 1^n.\) This proves Theorem 3.5 if \(M=1\), and *FH* is the bijection for \(M=1\).

*M*-version, \(M>1\), let \(\lambda \) have distinct parts. For \(\lambda =n\) a single part, define the

*M*-version by \(FH^M(n)= (M^k, 1^{n-kM})\), which has

*k*parts of size

*M*. Otherwise, \(\lambda \) has at least two parts, and

*kM*,

There are two cases. If \(\gamma \) has no parts of size *M*, define \(FH^M(\lambda )=\gamma \cup M^k\), so that \(FH^M(\lambda )\) is a partition with odd parts, exactly *k* parts of size *M*, and at most \(M-1\) 1’s.

If \(\gamma \) has \(r\ge 1\) parts of size *M*, change all of them to *rM* 1’s to obtain \(\gamma '\) with at least *M* 1’s. Then put \(FH^M(\lambda )=\gamma '\cup M^k,\) so that \(FH^M(\lambda )\) is a partition with odd parts, exactly *k* parts of size *M*, and at least *M* 1’s. \(\square \)

### Theorem 3.8

*M*, the number of partitions of

*n*into odd parts with exactly

*k*parts of size

*M*, is equal to the number of partitions \(\lambda \) of

*n*into distinct parts such that

- 1.
if \(\lambda \) has one part, then \(\lfloor n/M \rfloor =k,\)

- 2.
if \(\lambda \) has between two and \(M-1\) parts, then \(\lfloor (\lambda _1-\lambda _2-1)/M\rfloor =k,\)

- 3.
if \(\lambda \) has at least

*M*parts, then \(\lambda ^{St}\) has exactly*k**M*’s.

### Example 3.9

### 3.2 Göllnitz–Gordon Identities

*q*replaced by \(q^2,\)

*M*replaced by

*M*/ 2, and \(\alpha _j=q^{j^2}(-q;q^2)_j\) to obtain the next result.

### Corollary 3.10

*M*be a positive integer. Then

Here is the combinatorial restatement [6, Theorem 2] of the first Göllnitz–Gordon identity.

### Theorem 3.11

The number of partitions of *n* into parts congruent to 1, 4, or \(7\mod 8\) is equal to the number of partitions of *n* into parts whose difference is at least 2, and greater than 2 for consecutive even parts.

For the combinatorial version of Corollary 3.10, we need to recall why the sum side of (3.13.2) is the generating function for the restricted partitions with difference at least 2. In particular, we must identify what the denominator factor \(1-q\) represents in the sum side.

*j*parts. This is equivalent to showing that the generating function for \(\lambda ^*\) is

*j*parts, and the odd parts of \(\mu \) are distinct. The column read version \(\lambda ^*=\mu ^t\) can be built in the following way. Take arbitrary parts from sizes \(j, j-1, \ldots , 1\) with even multiplicity, whose generating function is \(1/(q^2;q^2)_j.\) The rows now have even length. Then, choose a subset of the odd integers \(1+0, 2+1, \ldots , j+(j-1).\) For each such odd part \(k+(k-1)\) add columns of length

*k*and \(k-1\). This keeps all rows even, except the

*k*-th row which is odd, and distinct.

### Theorem 3.12

*M*be a positive integer which is congruent to 1, 4 or \(7\mod 8.\) The number of partitions of \(n\ge 1\) into parts congruent to \(1,4{\text { or }}7\mod 8\) with exactly

*k*

*M*’s, is equal to the number of partitions \(\lambda \) of

*n*into parts whose difference is at least 2, and greater than 2 for consecutive even parts such that

- 1.
if \(\lambda \) has a single part, then \([n/M]=k,\)

- 2.if \(\lambda \) has at least two parts and the second part of \(\lambda \) is even, then$$\begin{aligned} \lfloor (\lambda _1-\lambda _2-3)/M\rfloor =k, \end{aligned}$$
- 3.if \(\lambda \) has at least two parts and the second part of \(\lambda \) is odd, then$$\begin{aligned} \lfloor (\lambda _1-\lambda _2-2)/M\rfloor =k. \end{aligned}$$

### Example 3.13

### Theorem 3.14

*M*be a positive integer which is congruent to 3, 4 or \(5\mod 8.\) The number of partitions of \(n\ge 1\) into parts congruent to \(3,4{\text { or }}5\mod 8\) with exactly

*k*

*M*’s, is equal to the number of partitions \(\lambda \) of

*n*into parts whose difference is at least 2, greater than 2 for consecutive even parts, smallest part at least 3, such that

- 1.
if \(\lambda \) has a single part, then \(n=Mk,\) or \(n=Mk+j\), \(3\le j\le M+2\), \(j\ne M\),

- 2.if \(\lambda \) has at least two parts and the second part of \(\lambda \) is even, then$$\begin{aligned} \lfloor (\lambda _1-\lambda _2-3)/M\rfloor =k, \end{aligned}$$
- 3.if \(\lambda \) has at least two parts and the second part of \(\lambda \) is odd, then$$\begin{aligned} \lfloor (\lambda _1-\lambda _2-2)/M\rfloor =k. \end{aligned}$$

## 4 An Andrews–Gordon Version

The Andrews–Gordon identities are

### Theorem 4.1

The Rogers–Ramanujan identities are the cases \(k=1\), \(a=0,1.\)

Because Theorem 4.1 has a multisum instead of a single sum, we cannot apply Proposition 3.1. Nonetheless, the same idea can be applied to obtain a marked version of Theorem 4.1.

### Theorem 4.2

*M*be any positive integer not congruent to 0, \(\pm (k+1-a)\) modulo \(2k+3.\) Then,

- 1.
for \(0\le a<k, \quad B=0, \quad A=q([M-1]_q+wq^{M-1})/(1-wq^M)\),

- 2.
for \(a=k, \quad B=n_1, \quad A=q^2([M-2]_q+wq^{M-2}+q^{M-1})/(1-wq^M).\)

For a combinatorial version of Theorem 4.2, we use Andrews’ Durfee dissections, and \((k+1,k+1-a)\)-admissible partitions, see [2].

### Definition 4.3

*k*be a positive integer and \(0\le a\le k.\) A partition \(\lambda \) is called \((k+1,k+1-a)\)-admissible if \(\lambda \) may be dissected by \(r\le k\) successive Durfee rectangles, moving down, of sizes

Note that \(r\le k-a\) is allowed, in which case all of the Durfee rectangles are squares. Also, the parts of \(\lambda \) to the right of the Durfee rectangles are not constrained, except at the last row of the non-square Durfee rectangle, where it is empty.

### Example 4.4

Suppose \(k=3\) and \(a=2\). Then \(\lambda =91\) is not (4, 2)-admissible: the Durfee square has size \(n_1=1\), but the next Durfee rectangle of size \(2\times 1\) does not exist, so the second part cannot be covered if \(r\ge 2.\)

Theorem 2 in [2] interprets Theorem 4.1.

### Proposition 4.5

The generating function for all partitions which are \((k+1,k+1-a)\)-admissible is given by the sum in Theorem 4.1.

*M*1’s in the columns to the right of the first Durfee rectangle are weighted by

*w*. These 1’s are again a difference in the first two parts of \(\lambda .\)

Putting these pieces together, the following result is a combinatorial restatement of Theorem 4.2 (Table 1).

### Theorem 4.6

*a*,

*k*,

*M*satisfying \(0\le a\le k\) and \(M\not \equiv 0, \pm (k+1-a)\mod 2k+3\). The number of partitions of

*n*into parts not congruent to 0, \(\pm (k+1-a)\mod 2k+3\) with exactly

*j*

*M*’s, is equal to the number of partitions \(\lambda \) of

*n*which are \((k+1,k+1-a)\)-admissible with \(r\le k\) Durfee rectangles of sizes

- 1.
if \(r=n_1=1,\) and \(0\le a<k\), then \(\lambda \) is a single part of size

*Mj*, \(Mj+1,\ldots , Mj+(M-1)\), - 2.
if \(r=n_1=1,\) and \(a=k\), then \(\lambda =(\lambda _1,1)\) has size

*Mj*, \(Mj+2,\ldots ,Mj+(M-1),\) or \(Mj+(M+1)\), - 3.
if \(n_1=1\) and \(r\ge 2,\) then \(\lfloor (\lambda _1-n_1)/M \rfloor =j,\)

- 4.
if \(n_1\ge 2,\) then \(\lfloor (\lambda _1-\lambda _2)/M \rfloor =j.\)

Theorem 4.6 when \(n=10, a=2, k=3, M=3\)

Partition without 2,7,9 | # of 3’s | (4,2)-Admissible partition | Value of |
---|---|---|---|

10 | 0 | 10 | 3 |

811 | 0 | 61, 111 | 1 |

64 | 0 | 421, 111 | 0 |

631 | 1 | 322, 111 | 0 |

61, 111 | 0 | 331, 111 | 0 |

55 | 0 | 811 | 2 |

541 | 0 | 6211 | 1 |

5311 | 1 | 5311 | 0 |

511, 111 | 0 | 5221 | 1 |

4411 | 0 | 22, 222 | 0 |

433 | 2 | 4411 | 0 |

43, 111 | 1 | 4321 | 0 |

4, 111, 111 | 0 | 82 | 2 |

3331 | 3 | 73 | 1 |

331, 111 | 2 | 64 | 0 |

31, 111, 111 | 1 | 55 | 0 |

1, 111, 111, 111 | 0 | 433 | 0 |

## 5 Shifting a Part

The weighted versions allow one to shift a part. For example in the first Rogers–Ramanujan identity, what happens if parts of size 11 are replaced by parts of size 28? All we need to do is to choose \(M=11\) and \(x=q^{17}\) in Theorem 1.1.

### Corollary 5.1

*M*be a positive integer which is congruent to 1 or 4 modulo 5. Let \(N>M\) be an integer not congruent to 1 or 4 modulo 5. The number of partitions of

*n*into parts congruent to 1 or 4 modulo 5, except

*M*, or parts of size

*N*, is equal to the number of partitions \(\lambda \) of

*n*with difference at least 2, such that

- 1.
\(\lambda \) has a single part, which is congruent to \(0,1,\ldots , {\text { or }}M-1\mod N,\)

- 2.
\(\lambda \) has at least two parts, and \(\lambda _1-\lambda _2-2\) is congruent to \(0,1,\ldots , {\text { or }}M-1\mod N.\)

### Example 5.2

### Theorem 5.3

*n*into parts from

*n*with difference at least 2 and

- 1.
if \(\lambda \) has a single part, then \(n\not \equiv 1\mod 3\),

- 2.
if \(\lambda \) has at least two parts, then \((\lambda _1-\lambda _2-2)\)\(\not \equiv 1\mod 3.\)

### Example 5.4

### Corollary 5.5

*M*be an odd positive integer. Let \(N>M\) be an even integer. The number of partitions of

*n*into odd parts except

*M*, or parts of size

*N*, is equal to the number of partitions \(\lambda \) of

*n*into distinct parts, such that

- 1.
\(\lambda \) has a single part, which is congruent to \(0,1,\ldots ,\) or \(M-1\mod N,\)

- 2.
\(\lambda \) has at least two parts, and \(\lambda _1-\lambda _2-1\) is congruent to \(0,1,\ldots ,\) or \(M-1\mod N.\)

### Example 5.6

## 6 Marking a Sum of Parts

One may ask if Theorems 1.1 and 2.1 have combinatorial interpretations without the modular conditions on *M*. The sum sides retain the interpretations given by Theorems 1.2 and 2.2 and are positive as a power series in *q* and *w*. It remains to understand what the product side represents as a generating function of partitions. We give in Proposition 6.4 a general positive combinatorial expansion for the product side. We call this “marking a sum of parts”.

*A*and

*B*are distinct integers congruent to 1 or \(4\mod 5.\) The quotient in the product side of Theorem 1.1

*A*or

*B*. The first term allows the number of

*B*’s to be at least as many as the number of

*A*’s. The second term allows the number of

*A*’s to be greater than the number of

*B*’s. The exponent of

*w*is the number of times a pair

*AB*appears in a partition. For example, if \(A=6\), \(B=4\), the partition (6, 6, 4, 4, 4, 4) contains 64 twice, along with two 4’s. We have found a prototypical result.

### Proposition 6.1

*AB*.

A more general statement holds for partitions other than \(M=A+B.\) To state this result, we need to define an analog of the multiplicity of a single part to a multiplicity of a partition. We again use the multiplicity notation for a partition, for example \((7^3,4^1, 2^3)\) denotes the partition (7, 7, 7, 4, 2, 2, 2).

### Definition 6.2

*k*times, \(k=E_\lambda (\mu ),\) if

### Example 6.3

Let \(\lambda =(6^1,4^2,1^1)\), \(\mu =(9^1,6^7, 4^5, 1^8).\) Then \(E_\lambda (\mu )=2\) but not 3 because \(\mu \) contains only five 4’s.

With this definition, Proposition 6.1 holds for any partition. We let \(|| \lambda ||\) denote the sum of the parts of \(\lambda .\)

### Proposition 6.4

The modular condition on the parts in Proposition 6.4 is irrelevant.

### Proposition 6.5

### Proof

*i*-th term of the sum represents partitions \(\mu =(B_1^{n_1}, B_2^{n_2},\ldots , B_k^{n_k})\), where

Adding back the multiples of \(\lambda \) by multiplying by \((1-wq^M)^{-1}\), and also the unused parts from \({\mathbb {A}}\), gives the result. \(\square \)

### Definition 6.6

Let \({\mathbb {A}}\) be a set of parts. If \(\lambda \) has parts from \({\mathbb {A}},\) let \(E_\lambda ^{{\mathbb {A}}}(n,k)\) be the number of partitions \(\mu \) of *n* with parts from \({\mathbb {A}}\) such that \(E_\lambda (\mu )=k.\)

### Corollary 6.7

*M*into parts from \({\mathbb {A}}.\) Then for all \(n,k\ge 0\)

Here are the promised versions of Theorems 1.2 and 2.2 when *M* does not satisfy the\(\mod 5\) condition.

### Corollary 6.8

Suppose that \(\lambda \) is a partition of *M* into parts congruent to 1 or \(4 \mod 5.\) Then, Theorem 1.2 holds if the number of partitions having *M* of multiplicity *k* is replaced by \(E_{\lambda }^{{\mathbb {A}}}(n,k)\), \({\mathbb {A}}=\{1,4,6,9,\ldots \}.\) Also, if \(\lambda \) is a partition of *M* into parts congruent to 2 or \(3 \mod 5,\) then Theorem 2.2 holds if the number of partitions having *M* of multiplicity *k* is replaced by \(E_{\lambda }^{{\mathbb {B}}}(n,k)\), \({\mathbb {B}}=\{2,3,7,8,\ldots \}.\)

### Example 6.9

One corollary of the Rogers–Ramanujan identities is that there are more partitions of *n* into parts congruent to 1 or \(4\mod 5\) than into parts congruent to 2 or \(3\mod 5.\) Kadell [8] gave an injection which proves this, and Berkovich–Garvan [3, Theorem 5.1] gave an injection for modulo 8. A general injection for finite products was given by Berkovich–Grizzell [4]. We can use Corollary 6.7, Theorems 1.1, and 2.1 to generalize this fact for the Rogers–Ramanujan identities.

### Theorem 6.10

### Proof

One could also apply Theorems 1.2 and 2.2 to obtain this result combinatorially. The single part case would be considered separately.

## 7 Remarks

- 1.
a single part \(\{M\},\)

- 2.
two parts \(\{2,M\},\)

- 3.
four parts \(\{2,3,7,8\}\).

*q*-analog of Euler’s odd=distinct theorem [10, Theorem 1] is the following. Let

*q*be a positive integer. The number of partitions of

*N*into

*q*-odd parts \([2k+1]_q\) is equal to the the number of partitions of

*N*into parts \([m]_q\) whose multiplicity is \(\le q^m\). A generating function identity equivalent to this result is

Given \(\lambda \) and \(\mu \), \(E_\lambda (\mu )\) is an integer which counts the number of \(\lambda \)’s in \(\mu \). One could imagine defining instead a rational value for this “multiplicity”.

## Notes

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