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Annals of Combinatorics

, Volume 23, Issue 3–4, pp 935–951 | Cite as

Marking and Shifting a Part in Partitions

  • Kathleen O’Hara
  • Dennis StantonEmail author
Article
  • 33 Downloads

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

Many integer partition theorems can be restated as an analytic identity, as a sum equal to a product. One such example is the first Rogers–Ramanujan identity
$$\begin{aligned} \frac{1}{\prod _{k=0}^\infty (1-q^{5k+1})(1-q^{5k+4})}= 1+\sum _{k=1}^{\infty } q^{k^2}\frac{1}{(1-q)(1-q^2)\cdots (1-q^k)}.\nonumber \\ \end{aligned}$$
(1.1)
MacMahon’s combinatorial version of (1.1) uses integer partitions. The left side is the generating function for all partitions whose parts are congruent to 1 or \(4\mod 5.\) The factor \(1/(1-q^9)\) on the left side allows an arbitrary number of 9’s in an integer partition. If we “mark” or weight the 9 by a w, the factor \(1/(1-q^9)\) is replaced by
$$\begin{aligned} \frac{1}{1-wq^9}. \end{aligned}$$
One may ask how the right side is modified upon marking a part, and whether a refined combinatorial interpretation exists.

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

Let \(M\ge 1\) be any integer congruent to 1 or \(4\mod 5\). Then
$$\begin{aligned} \begin{aligned}&\frac{1-q^M}{1-wq^M}\frac{1}{\prod _{k=0}^\infty (1-q^{5k+1})(1-q^{5k+4})}= 1+q\frac{1+q+\cdots +q^{M-2}+wq^{M-1}}{1-wq^M}\\&\quad +\sum _{k=2}^{\infty } q^{k^2} \frac{1+q+\cdots +q^{M-1}}{1-wq^M}\frac{1}{(1-q^2)(1-q^3)\cdots (1-q^k)}. \end{aligned} \end{aligned}$$

Here is a combinatorial version of Theorem 1.1.

Theorem 1.2

Let 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 kM’s is equal to the number of partitions \(\lambda \) of n with difference at least 2 and
  1. 1.

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

     
  2. 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.

We shall also consider “shifting” a part, for example replacing all 9’s by 22’s in (1.1). This is replacing the factor
$$\begin{aligned} \frac{1}{1-q^9} \quad {\text {by}} \quad \frac{1}{1-q^{22}}. \end{aligned}$$
We shall see that the set of partitions enumerated by the sum side is an explicit subset of the partitions in the original identity.

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.

We use the standard notation,
$$\begin{aligned} (A;q)_k=\prod _{j=0}^{k-1}(1-Aq^j), \quad [M]_q=\frac{1-q^M}{1-q}. \end{aligned}$$
If the base 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

Let \(M\ge 2\) be any integer congruent to 2 or \(3\mod 5\). Then
$$\begin{aligned} \begin{aligned} \frac{1-q^M}{1-wq^M}\frac{1}{(q^2;q^5)_\infty (q^3;q^5)_\infty }&= 1+q^2\frac{[M-2]_q+wq^{M-2}+q^{M-1}}{1-wq^M}\\&\quad +\sum _{k=2}^{\infty } q^{k^2+k} \frac{[M]_q}{1-wq^M}\frac{1}{(q^2;q)_{k-1}}. \end{aligned} \end{aligned}$$

Here is a combinatorial version of Theorem 2.1.

Theorem 2.2

Let 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 kM’s is equal to the number of partitions \(\lambda \) of n with difference at least 2, no 1’s, and
  1. 1.

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

     
  2. 2.

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

     

Proof

We simultaneously prove Theorems 1.2 and 2.2. We need to understand the combinatorics of the replacement in the k-th term on the sum side
$$\begin{aligned} \frac{1}{1-q}\rightarrow \frac{[M]_q}{1-wq^M}=\sum _{p=0}^\infty q^p w^{\lfloor p/M\rfloor }. \end{aligned}$$
(2.1)
In the classical Rogers–Ramanujan identities, the factor \(1/(1-q)\) represents the difference in the first two parts after the double staircase has been removed. This is the second case of each theorem. \(\square \)

Example 2.3

Let \(k=2\), \(M=7\), and \(n=22.\) The equinumerous sets of partitions for Theorem 2.2 are
$$\begin{aligned} \{(8,7,7), (7,7,3,3,2), (7,7,2,2,2,2) \} \leftrightarrow \{ (22), (20,2), (19,3)\}. \end{aligned}$$
Equivalent combinatorial versions of Theorems 1.2 and 2.2 may be given (see [9, Theorem 2, Theorem 3]). This time, the terms \(k\ge M\) of the sum side are considered, and the replacement considered is
$$\begin{aligned} \frac{1}{1-q^M}\rightarrow \frac{1}{1-wq^M}, \end{aligned}$$
namely, the part 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

Let 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 kM’s is equal to the number of partitions \(\lambda \) of n with difference at least 2 and
  1. 1.

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

     
  2. 2.

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

     
  3. 3.

    if \(\lambda \) has at least M parts, then \(\lambda ^*\) has exactly kM’s.

     

Example 2.6

Let \(k=2\), \(M=4\), and \(n=24.\) The equinumerous sets of partitions for Theorem 2.5 are
$$\begin{aligned} \begin{aligned}&\{(16,4^2), (14,4^2,1^2), (11,4^2,1^5), (9,6,4^2,1), (9,4^2,1^7), (6,6,4^2,1^4),\\&\quad (6,4^2,1^{10}),(4^2,1^{16})\}\leftrightarrow \\&\quad \{(9,7,5,3), (18,5,1), (17,6,1), (17,5,2), (16,6,2), (16,5,3), (17,7), (18,6)\}. \end{aligned} \end{aligned}$$

Theorem 2.7

Let 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 kM’s is equal to the number of partitions \(\lambda \) of n with difference at least 2, no 1’s and
  1. 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. 2.

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

     
  3. 3.

    if \(\lambda \) has at least M parts, then \(\lambda ^{**}\) has exactly kM’s.

     

3 A General Expansion

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

Many partition identities have a sum side of the form
$$\begin{aligned} \sum _{j=0}^\infty \frac{\alpha _j}{(q;q)_j}, \end{aligned}$$
where \(\alpha _j\) has non-negative coefficients as a power series in q.
These include
  1. 1.

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

     
  2. 2.

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

     
  3. 3.

    the Göllnitz–Gordon identities, q replaced by \(q^2\), \(\alpha _j=q^{j^2}(-q;q^2)_j,\)

     
  4. 4.

    all partitions by the largest part, \(\alpha _j=q^j,\)

     
  5. 5.

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

     
A part of size M may be marked in general using the next proposition.

Proposition 3.1

For any positive integer M, if \(\alpha _0=1\),
$$\begin{aligned} \frac{1-q^M}{1-wq^M}\sum _{j=0}^\infty \frac{\alpha _j}{(q;q)_j}= 1+\frac{\alpha _1[M]_q-q^M+wq^M}{1-wq^M}+\sum _{j=2}^\infty \frac{[M]_q}{1-wq^M}\frac{\alpha _j}{(q^2;q)_{j-1}}. \end{aligned}$$
As long as \(\alpha _1\) has the property that
$$\begin{aligned} \alpha _1 [M]_q-q^M \end{aligned}$$
is a positive power series in q, the right side has a combinatorial interpretation.
There are two possible elementary combinatorial interpretations. For any \(j\ge 2,\) the factor
$$\begin{aligned} \frac{[M]_q}{1-wq^M}=\sum _{p=0}^\infty q^p w^{[p/M]} \end{aligned}$$
replaces \(1/(1-q),\) which accounts for parts of size 1 in a partition. This is a weighted form of the number of 1’s.
The second interpretation holds for terms with \(j\ge M.\) Here
$$\begin{aligned}&\frac{[M]_q}{1-wq^M}\frac{1}{(q^2;q)_{j-1}}\\&\quad =\frac{1}{(1-q)\cdots (1-q^{M-1})(1-wq^M)(1-q^{M+1})\cdots (1-q^j)}. \end{aligned}$$
In this case, the part of size M is marked by w.
For a particular combinatorial application of Proposition 3.1, one must realize what the denominator factors \((1-q)\) and \((1-q^M)\) represent on the sum side. For example, in the first Rogers–Ramanujan identity, these factors account for 1’s and M’s in \(\lambda ^*.\) Since
$$\begin{aligned} (\#1's {\text { in }}\lambda ^*)=\lambda _1-\lambda _2-2, \end{aligned}$$
the two interpretations are Theorems 1.2 and 2.5.

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

For any odd positive integer M,
$$\begin{aligned} \begin{aligned}&\frac{1}{(1-q)(1-q^3)\cdots (1-q^{M-2})(1-wq^M)(1-q^{M+2})\cdots }\\&\quad = 1+\frac{q+q^2+\cdots +q^{M-1}+wq^M}{1-wq^M}+\sum _{j=2}^\infty \frac{q^{\left( {\begin{array}{c}j+1\\ 2\end{array}}\right) }}{(q^2;q)_{j-1}}\frac{[M]_q}{1-wq^M}. \end{aligned} \end{aligned}$$

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

For any odd positive integer 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. 1.

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

     
  2. 2.

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

     

Example 3.6

Let \(k=2\), \(M=5\), and \(n=18.\) The equinumerous sets of partitions for Theorem 3.5 are
$$\begin{aligned} \begin{aligned}&\{(7,5,5,1),(5,5,3,3,1,1),(5,5,3,1^5), (5,5,1^8)\}\\&\quad \leftrightarrow \{(16,2),(15,3), (15,2,1), (14,3,1)\}. \end{aligned} \end{aligned}$$

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 \(M=1\).

For the 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
$$\begin{aligned} kM\le \lambda _1-\lambda _2-1\le (k+1)M-1. \end{aligned}$$
Let \(\theta \) be the partition with distinct parts where \(\lambda _1\) has been reduced by kM
$$\begin{aligned} 0\le \theta _1-\theta _2-1\le M-1. \end{aligned}$$
Finally put \(\gamma =FH(\theta ),\) and note that \(\gamma \) has at most \(M-1\) 1’s.

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

For any odd positive integer 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. 1.

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

     
  2. 2.

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

     
  3. 3.

    if \(\lambda \) has at least M parts, then \(\lambda ^{St}\) has exactly kM’s.

     

Example 3.9

Let \(k=2\), \(M=3\), and \(n=18.\) The equinumerous sets of partitions for Theorem 3.8 are
$$\begin{aligned} \begin{aligned}&\{(11,3^2,1),(9,3^2,1^3), (7,5,3^2), (7,3^2,1^5),(5^2,3^2,1^2),(5,3^2,1^7), (3^2,1^{12})\}\\&\quad \leftrightarrow \{(7,6,4,1),(8,5,4,1),(11,4,3),(10,5,3),(9,6,3),(8,7,3),(13,5)\}. \end{aligned} \end{aligned}$$

3.2 Göllnitz–Gordon Identities

The Göllnitz–Gordon identities are (see [1, 5, 6])
$$\begin{aligned} \sum _{n=0}^\infty q^{n^2}\frac{(-q;q^2)_n}{(q^2;q^2)_n}= & {} \frac{1}{(q;q^8)_\infty (q^4;q^8)_\infty (q^7;q^8)_\infty }, \end{aligned}$$
(3.1)
$$\begin{aligned} \sum _{n=0}^\infty q^{n^2+2n}\frac{(-q;q^2)_n}{(q^2;q^2)_n}= & {} \frac{1}{(q^3;q^8)_\infty (q^4;q^8)_\infty (q^5;q^8)_\infty }. \end{aligned}$$
(3.2)
We apply Proposition 3.1 with 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

Let M be a positive integer. Then
$$\begin{aligned} \begin{aligned}&\frac{1-q^M}{1-wq^M}\frac{1}{(q;q^8)_\infty (q^4;q^8)_\infty (q^7;q^8)_\infty }\\&\quad =1+\frac{q[M-1]_q+wq^M}{1-wq^M} +\sum _{j=2}^\infty q^{j^2}\frac{[M]_{q}}{1-wq^M}\frac{(-q^3;q^2)_{j-1}}{(q^4;q^2)_{j-1}}. \end{aligned} \end{aligned}$$
We used
$$\begin{aligned} \frac{q(1+q)[M/2]_{q^2}-q^M+wq^M}{1-wq^M}=\frac{q[M-1]_q+wq^M}{1-wq^M} \end{aligned}$$
to simplify the second term in the sum in Corollary 3.10. Note that the numerator has positive coefficients, and thus a simple combinatorial interpretation.

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.

Suppose \(\lambda \) is such a partition with j parts. This is equivalent to showing that the generating function for \(\lambda ^*\) is
$$\begin{aligned} \frac{(-q;q^2)_j}{(q^2;q^2)_j}=\frac{1+q}{1-q^2}\frac{(-q^3;q^2)_{j-1}}{(q^4;q^2)_{j-1}}. \end{aligned}$$
(3.3)
The partition \(\mu =\lambda -(2j-1,2j-3,\ldots , 1)\) has at most 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.
We see that the factor \((1+q)/(1-q^2)=1/(1-q)\) in (3.3) accounts for 1’s in \(\lambda ^*.\) In Corollary 3.10, this quotient is replaced by
$$\begin{aligned} \frac{1+q}{1-q^2}\rightarrow \frac{[M]_q}{1-wq^M}=\sum _{p=0}^\infty q^p w^{[p/M]}. \end{aligned}$$
There is one final opportunity for a 1 to appear in \(\lambda ^*\): when \(3=2+1\) is chosen as an odd part. This occurs only when the second part of \(\lambda \) is even.

Theorem 3.12

Let 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 kM’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. 1.

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

     
  2. 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. 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

Let \(k=3\), \(M=7\), and \(n=31.\) The equinumerous sets of partitions for Theorem 3.12 are
$$\begin{aligned} \begin{aligned}&\{(9,7,7,7,1), (7,7,7,4,4,1,1), (7,7,7,4,1^6), (7,7,7,1^{10})\}\\&\quad \leftrightarrow \{(30,1), (29,2),(28,3), (27,3,1)\}. \end{aligned} \end{aligned}$$
Note that \(\lambda =(27,4)\) is not allowed because the second part of \(\lambda \) is even.
For the second Göllnitz–Gordon identity, the version of Corollary 3.10 is
$$\begin{aligned} \begin{aligned}&\frac{1-q^M}{1-wq^M}\frac{1}{(q^3;q^8)_\infty (q^4;q^8)_\infty (q^5;q^8)_\infty }\\&\quad =1+\frac{q^3+\cdots +q^{M-1}+wq^M+q^{M+1}+q^{M+2}}{1-wq^M}\\&\qquad +\sum _{j=2}^\infty q^{j^2+2j}\frac{[M]_{q}}{1-wq^M}\frac{(-q^3;q^2)_{j-1}}{(q^4;q^2)_{j-1}}. \end{aligned} \end{aligned}$$
(3.4)
Here is the combinatorial refinement of [6, Theorem 3].

Theorem 3.14

Let 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 kM’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. 1.

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

     
  2. 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. 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

If \(0\le a \le k\), then
$$\begin{aligned} \frac{(q^{k+1-a},q^{k+2+a},q^{2k+3};q)_\infty }{(q;q)_\infty }= \sum _{n_1\ge n_{2} \ge \cdots \ge n_k\ge 0} \frac{q^{n_1^2+n_2^2+\cdots +n_k^2+n_{k+1-a}+\cdots +n_k}}{(q)_{n_1-n_2}\cdots (q)_{n_{k-1}-n_k}(q)_{n_k}}. \end{aligned}$$

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.

Let \(F_k^a\) denote the right-side multisum of Theorem 4.1 for \(0\le a\le k\), and let \(F_k^a=F_k^0\) for \(a<0.\) So we have
$$\begin{aligned} F_k^a=F_{k-1}^{a-1}+\sum _{n_1\ge n_{2} \ge \cdots \ge n_k\ge 1} \frac{q^{n_1^2+n_2^2+\cdots +n_k^2+n_{k+1-a}+\cdots +n_k}}{(q)_{n_1-n_2}\cdots (q)_{n_{k-1}-n_k}(q)_{n_k}}. \end{aligned}$$
Multiplying by \(\frac{1-q^M}{1-wq^M}\) yields
$$\begin{aligned}&\frac{1-q^M}{1-wq^M} F_k^a=\frac{1-q^M}{1-wq^M} F_{k-1}^{a-1}\\&\quad +\sum _{n_1\ge n_{2} \ge \cdots \ge n_k\ge 1} \frac{q^{n_1^2+n_2^2+\cdots +n_k^2+n_{k+1-a}+\cdots +n_k}}{(q)_{n_1-n_2}\cdots (q)_{n_{k-1}-n_k}(q^2;q)_{n_k-1}}\frac{[M]_q}{1-wq^M}, \end{aligned}$$
which, upon iterating, is the following weighted version of the Andrews–Gordon identities.

Theorem 4.2

For \(0\le a\le k\), let M be any positive integer not congruent to 0, \(\pm (k+1-a)\) modulo \(2k+3.\) Then,
$$\begin{aligned} \begin{aligned}&\frac{1-q^M}{1-wq^M}\frac{(q^{k+1-a},q^{k+2+a},q^{2k+3};q)_\infty }{(q;q)_\infty } =1+A+ \sum _{n_1=2}^\infty \frac{q^{n_1^2+B}}{(q^2;q)_{n_1-1}}\frac{[M]_q}{1-wq^M}\\&\quad +\sum _{r=2}^k\sum _{n_1\ge n_{2} \ge \cdots \ge n_r\ge 1} \frac{q^{n_1^2+n_2^2+\cdots +n_r^2+n_{k+1-a}+\cdots +n_r}}{(q)_{n_1-n_2}\cdots (q)_{n_{r-1}-n_r}(q^2;q)_{n_r-1}}\frac{[M]_q}{1-wq^M}, \end{aligned} \end{aligned}$$
where
  1. 1.

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

     
  2. 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

Let 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
$$\begin{aligned} n_1\times n_1,\ldots , n_{k-a}\times n_{k-a},\ (n_{k-a+1}+1)\times n_{k-a}, \ldots ,(n_r+1) \times n_r \end{aligned}$$
such that the \((n_1+n_2+\cdots + n_{k-a+i}+i)\)-th part of \(\lambda \) is \( n_{k-a+i},\) for \(1\le i\le r-(k-a).\)

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.

We need to understand the replacement
$$\begin{aligned} \frac{1}{(q)_{n_r}}=\frac{1}{(1-q) (q^2;q)_{n_r-1}} \rightarrow \frac{1}{(q^2;q)_{n_r-1}}\frac{[M]_q}{1-wq^M} \end{aligned}$$
in the factor \((q)_{n_r}\) to give a combinatorial version of Theorem 4.2.
First, we recall [2] that if the sizes of the Durfee rectangles are fixed by \(n_1,n_2,\ldots ,n_r,\) then the generating function for the partitions which have this Durfee dissection is
$$\begin{aligned} \frac{1}{(q)_{n_1}}\prod _{j=1}^{r-1} {\left[ {n_j \atop n_{j+1}} \right] }_q= \frac{1}{(q)_{n_r}}\prod _{j=1}^{r-1}\frac{1}{(q)_{n_j-n_{j+1}}}. \end{aligned}$$
(A simple bijection for this fact is given in [7].) Upon multiplying by
$$\begin{aligned} \frac{1-q^M}{1-wq^M}, \end{aligned}$$
we have
$$\begin{aligned} \frac{[M]_q}{1-wq^M}\frac{1}{(q^2;q)_{n_1-1}}\prod _{j=1}^{r-1} {\left[ {n_j \atop n_{j+1}} \right] }_q= \frac{[M]_q}{1-wq^M}\frac{1}{(q^2;q)_{n_r-1}}\prod _{j=1}^{r-1}\frac{1}{(q)_{n_j-n_{j+1}}}. \end{aligned}$$
Consider the factor \(1/(q)_{n_1},\) which accounts for the portion of the partition to the right of the first Durfee rectangle of \(\lambda .\) In this factor, we are replacing
$$\begin{aligned} \frac{1}{1-q}\rightarrow \frac{[M]_q}{1-wq^M}. \end{aligned}$$
As before, the 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

Fix integers akM 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 jM’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
$$\begin{aligned} n_1\times n_1,\ldots , n_{k-a}\times n_{k-a}, (n_{k-a+1}+1)\times n_{k-a}, \ldots ,(n_r+1) \times n_r \end{aligned}$$
of the following form:
  1. 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. 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. 3.

    if \(n_1=1\) and \(r\ge 2,\) then \(\lfloor (\lambda _1-n_1)/M \rfloor =j,\)

     
  4. 4.

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

     
Table 1

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 j

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

Let 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. 1.

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

     
  2. 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

Let \(N=8\), \(M=4\), and \(n=9.\) The equinumerous sets of partitions for Corollary 5.1 are
$$\begin{aligned} \{(9),(6,1,1,1),(8,1), (1^9)\}\leftrightarrow \{(9),(6,3), (7,2), (5,3,1)\}. \end{aligned}$$
A related example occurs when two parts are shifted: 1 and 4 are replaced by 2 and 3. The appropriate identity is
$$\begin{aligned} \frac{1}{(1-q^2)(1-q^3)(q^6;q^5)_\infty (q^9;q^5)_\infty } = 1+\frac{q^2(1+q)}{1-q^3}+\sum _{k=2}^\infty \frac{q^{k^2}}{(q^2;q)_{k-1}}\frac{1+q^2}{1-q^3}.\nonumber \\ \end{aligned}$$
(5.1)

Theorem 5.3

The number of partitions of n into parts from
$$\begin{aligned} \{2,3,5k+1,5k+4: k\ge 1\} \end{aligned}$$
is equal to the number of partitions \(\lambda \) of n with difference at least 2 and
  1. 1.

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

     
  2. 2.

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

     

Example 5.4

Let \(n=13\). The two equinumerous sets of partitions in Theorem 5.3 are
$$\begin{aligned} \begin{aligned}&\{(11,2), (9,2,2), (6,3,2,2), (3,2,2,2,2,2), (3,3,3,2,2)\}\\&\quad \leftrightarrow \{(12,1), (10,3), (9,4), (8,4,1), (7,5,1)\}. \end{aligned} \end{aligned}$$
The possible partitions with difference at least 2
$$\begin{aligned} \{(13), (11,2), (8,5), (9,3,1), (7,4,2)\} \end{aligned}$$
are disallowed.

Corollary 5.5

Let 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. 1.

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

     
  2. 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

If \(N=8\), \(M=3\) and \(n=9\), the equinumerous sets in Corollary 5.5 are
$$\begin{aligned} \{(9), (8,1), (7,1,1), (5,1^4), (1^9)\}\leftrightarrow \{(9), (5,4), (6,3), (5,3,1), (4,3,2)\}. \end{aligned}$$

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”.

As an example suppose that \(M=A+B\) is a sum of two parts, where A and B are distinct integers congruent to 1 or \(4\mod 5.\) The quotient in the product side of Theorem 1.1
$$\begin{aligned} \frac{1-q^{A+B}}{1-wq^{A+B}}\frac{1}{(1-q^A)(1-q^B)}= \frac{1}{(1-q^B)(1-wq^{A+B})}+\frac{q^A}{(1-q^A)(1-wq^{A+B})} \end{aligned}$$
is a generating function for partitions with parts 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

Let \(M=A+B\) for some \(A,B\equiv 1,4 \mod 5, A\ne B.\) Then
$$\begin{aligned} \frac{1-q^M}{1-wq^M} \frac{1}{(q;q^5)_\infty (q^4;q^5)_\infty } \end{aligned}$$
is the generating function for all partitions \(\mu \) with parts \(\equiv 1,4\mod 5\) by the number of occurrences of the pair 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

Let \(\lambda =(A_1^{m_1},\ldots , A_k^{m_k})\) be a partition. We say \(\lambda \) is inside \(\mu \)k times, \(k=E_\lambda (\mu ),\) if
$$\begin{aligned} k=\max \{j: j\ge 0, \mu {\text { contains at least }} jm_s {\text { parts of size }}A_s {\text { for all }}s\}. \end{aligned}$$

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

Let \(\lambda \vdash M\) be a fixed partition into parts congruent to 1 or \(4\mod 5.\) Then
$$\begin{aligned} \frac{1-q^M}{1-wq^M} \frac{1}{(q;q^5)_\infty (q^4;q^5)_\infty } \end{aligned}$$
is the generating function for all partitions \(\mu \) into parts congruent to 1 or \(4\mod 5,\)
$$\begin{aligned} \sum _{\mu } q^{||\mu ||} w^{E_\lambda (\mu )}, \end{aligned}$$
where \(E_\lambda (\mu )\) is the number of times \(\lambda \) appears in \(\mu .\)

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

Proposition 6.5

Let \({\mathbb {A}}=\{A_1,A_2,\ldots \}\) be any set of positive integers. Suppose that \(\lambda =(B_1^{m_1},\ldots , B_k^{m_k})\) is a partition whose parts come from \({\mathbb {A}}\) and \(M=\sum _{i=1}^k m_i B_i.\) Then
$$\begin{aligned} \frac{1-q^M}{1-wq^M}\prod _{i=1}^\infty (1-q^{A_i})^{-1} \end{aligned}$$
is the generating function for all partitions \(\mu \) with parts from \({\mathbb {A}}\)
$$\begin{aligned} \sum _{\mu } q^{||\mu ||} w^{E_\lambda (\mu )}. \end{aligned}$$

Proof

We start with the telescoping sum
$$\begin{aligned} 1-q^M= 1-q^{m_1B_1}+q^{m_1B_1}(1-q^{m_2B_2})+\cdots +q^{\sum _{i=1}^{k-1}m_iB_i}(1-q^{m_kB_k}), \end{aligned}$$
which implies
$$\begin{aligned} \begin{aligned}&(1-q^M)\prod _{i=1}^k (1-q^{B_i})^{-1}\\&\quad =\sum _{i=1}^k q^{m_1B_1+\cdots +m_{i-1}B_{i-1}} \prod _{j=1}^{i-1} (1-q^{B_j})^{-1} \frac{1-q^{m_iB_i}}{1-q^{B_i}} \prod _{j=i+1}^{k} (1-q^{B_j})^{-1}. \end{aligned} \end{aligned}$$
(6.1)
We see that (6.1) is the generating function for partitions \(\mu \) with parts from \(\{B_1,B_2,\ldots , B_k\}\) such that \(E_\lambda (\mu )=0.\) The i-th term of the sum represents partitions \(\mu =(B_1^{n_1}, B_2^{n_2},\ldots , B_k^{n_k})\), where
$$\begin{aligned} n_1\ge m_1, \ n_2\ge m_2,\ldots , n_{i-1}\ge m_{i-1}, \ n_i< m_i. \end{aligned}$$
These disjoint sets cover all \(\mu \) with \(E_\lambda (\mu )=0.\)

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

For any set of part sizes \({\mathbb {A}}\), let \(\lambda _1\) and \(\lambda _2\) be two partitions of M into parts from \({\mathbb {A}}.\) Then for all \(n,k\ge 0\)
$$\begin{aligned} E_{\lambda _1}^{{\mathbb {A}}}(n,k)=E_{\lambda _2}^{{\mathbb {A}}}(n,k). \end{aligned}$$

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

Let \(\lambda =(6,1)\), \(M=7\), and \(n=17.\) The equinumerous sets of partitions for Corollary 6.8 are
$$\begin{aligned} \begin{aligned}&\{ (9,6,1,1), (6,6,4,1), (6,4,4,1,1,1), (6,4,1^7), (6,1^{11})\}\\&\quad \leftrightarrow \{ (16,1), (15,2), (14,3), (13,4), (13,3,1)\}. \end{aligned} \end{aligned}$$

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

Let
$$\begin{aligned} {\mathbb {A}}=\{ 5k+1, 5k+4: k\ge 0\}, \quad {\mathbb {B}}=\{ 5k+2, 5k+3: k\ge 0\}. \end{aligned}$$
Fix partitions \(\lambda \vdash M\) and \(\theta \vdash M\), \(M\ge 3\), with parts from \({\mathbb {A}}\) and \({\mathbb {B}}\), respectively. Then for all \(n,k\ge 0\)
$$\begin{aligned} E_{\theta }^{{\mathbb {B}}}(n,k)\le E_{\lambda }^{{\mathbb {A}}}(n,k). \end{aligned}$$

Proof

By Corollary 6.7, Theorems 1.1, and 2.1 we have
$$\begin{aligned} \begin{aligned}&\sum _{n=0}^\infty \sum _{k=0}^\infty q^n w^k ( E_{\lambda }^{{\mathbb {A}}}(n,k)-E_{\theta }^{{\mathbb {B}}}(n,k))= \frac{q-q^{M+1}}{1-wq^M}+\sum _{k=2}^\infty q^{k^2} \frac{[M]_q}{1-wq^M}\frac{1}{(q^2;q)_{k-2}}. \end{aligned} \end{aligned}$$
All terms are positive except for the first term. If we add the \(k=2\) term to the first term, we have
$$\begin{aligned} \frac{q-q^{M+1}+q^4[M]_q}{1-wq^M}, \end{aligned}$$
whose numerator is positive for \(M\ge 3.\)\(\square \)

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

All of parts 1, \(2,\ldots \) may be simultaneously marked for all partitions by the largest part. The corresponding identity is
$$\begin{aligned} \frac{1}{\prod _{k=1}^\infty (1-x_k q^k)}=1+\sum _{j=1}^\infty \frac{x_j q^j}{\prod _{k=1}^j (1-x_k q^k)}. \end{aligned}$$
A corresponding rational function identity for marking \(1,2,\ldots ,n\) is
$$\begin{aligned} \frac{1}{\prod _{k=1}^n (1-x_k q^k)}\left( \sum _{j=0}^n \frac{q^{j}}{(q)_j}\right) = \frac{1}{(q)_n}\left( 1+\sum _{j=1}^n \frac{x_j q^j}{\prod _{k=1}^j (1-x_k q^k)}\right) . \end{aligned}$$
We do not have such general marked versions for the Rogers–Ramanujan identities, which would be equivalent to bijections. There are partial results. In Ref. [9] marked versions of the second Rogers–Ramanujan identity are given for
  1. 1.

    a single part \(\{M\},\)

     
  2. 2.

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

     
  3. 3.

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

     
We do not have a general version of Proposition 3.1 which gives the last marked version.
A 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
$$\begin{aligned} \prod _{n=0}^\infty \frac{1}{1-t^{[2n+1]_q}}= 1+\sum _{m=1}^\infty t^{[m]_q} \frac{1-t^{q^m[m]_q}}{1-t^{[m]_q}}\prod _{k=1}^{m-1} \frac{1-t^{(q^k+1)[k]_q}}{1-t^{[k]_q}}. \end{aligned}$$
We do not know how to perturb this identity to mark a part.

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

References

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.BlacksburgUSA
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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