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Combinatorial approach to Andrews-Gordon and Bressoud type identities

Jehanne Dousse, Frédéric Jouhet, Isaac Konan

Abstract

We provide combinatorial tools inspired by work of Warnaar to give combinatorial interpretations of the sum sides of the Andrews-Gordon and Bressoud identities. More precisely, we give an explicit weight- and length-preserving bijection between sets related to integer partitions, which provides these interpretations. In passing, we discover the $q$-series version of an identity of Kurşungöz, similar to the Bressoud identity but with opposite parity conditions, which we prove combinatorially using the classical Bressoud identity and our bijection. We also use this bijection to prove combinatorially many identities, some known and other new, of the Andrews-Gordon and Bressoud type.

Combinatorial approach to Andrews-Gordon and Bressoud type identities

Abstract

We provide combinatorial tools inspired by work of Warnaar to give combinatorial interpretations of the sum sides of the Andrews-Gordon and Bressoud identities. More precisely, we give an explicit weight- and length-preserving bijection between sets related to integer partitions, which provides these interpretations. In passing, we discover the -series version of an identity of Kurşungöz, similar to the Bressoud identity but with opposite parity conditions, which we prove combinatorially using the classical Bressoud identity and our bijection. We also use this bijection to prove combinatorially many identities, some known and other new, of the Andrews-Gordon and Bressoud type.
Paper Structure (16 sections, 35 theorems, 70 equations, 6 figures)

This paper contains 16 sections, 35 theorems, 70 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The setup for our combinatorial approach
  3. Combinatorial description of the right-hand (product or sum of products) sides
  4. Combinatorial description of the left-hand (multisum) sides
  5. Proof of Theorem \ref{['th:bij']}
  6. The case $r=2$
  7. The map $\Lambda:\mathcal{P}_{r}\to\mathcal{A}_{r}$
  8. The map $\Gamma:\mathcal{A}_{r}\to\mathcal{P}_{r}$
  9. $\Gamma\circ\Lambda$ is the identity on $\mathcal{P}_r$
  10. $\Lambda\circ \Gamma$ is the identity on $\mathcal{A}_{r}$
  11. Proof of Corollaries \ref{['coro:GF']}--\ref{['coro:list']}
  12. Maps induced by $\Lambda$
  13. Maps induced by $\Gamma$
  14. Proof of Corollary \ref{['coro:GF']}
  15. Proof of Corollary \ref{['coro:newbressoud']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $i=1$ or $2$. For all non-negative integers $n$, the number of partitions of $n$ such that the difference between two consecutive parts is at least $2$ and the part $1$ appears at most $i-1$ times is equal to the number of partitions of $n$ into parts congruent to $\pm (2+i) \mod 5.$

Figures (6)

  • Figure 1: The maps in terms of parts.
  • Figure 2: The map $\Lambda$ in terms of frequencies.
  • Figure 3: The map $\Gamma$ in terms of frequencies.
  • Figure 4: An example when $r=8$.
  • Figure 5: Effects of $\Lambda$ on the multiplicity sequence from step $u+1$ to step $u$.
  • ...and 1 more figures

Theorems & Definitions (69)

  • Theorem 1.1: Rogers--Ramanujan identities, partition version
  • Theorem 1.2: Gordon's identities
  • Theorem 1.3: Andrews--Gordon identities
  • Theorem 1.4: Bressoud's identities
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7: Bijection
  • Corollary 1.8: Sum sides of the Andrews--Gordon and Bressoud identities
  • Corollary 1.9: Kurşungöz identities, new multisum
  • Corollary 1.10: Andrews--Gordon and Bressoud type identities
  • ...and 59 more