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A MacMahon Analysis View of Cylindric Partitions

Runqiao Li, Ali K. Uncu

TL;DR

This work applies MacMahon partition analysis and the Omega operator to study cylindric partitions with two-element profiles, deriving explicit refined generating functions and recurrences for CP_{(c1,c2)}(n) and their bilateral, manifestly positive refinements P_{(c1,c2)}(n). It connects these finite-form identities to classical Rogers–Ramanujan–type theorems and their polynomial refinements (Andrews–Gordon, Bressoud, Foda–Quano) and shows how limits yield Borodin’s product formulas, while providing several explicit bilateral sums with nonnegative coefficients. The authors prove numerous cases (notably k=2,3,4) and formulate conjectures for infinite hierarchies that they partially realize via Bailey machinery, offering a program to establish broad families of polynomial identities. The results illuminate deep links between cylindric partitions, q-series, and Bailey-type techniques, and they propose concrete pathways to extend polynomial refinements and hierarchies beyond the proven cases.

Abstract

We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analysis and then solving them. We also note some q-series identities related to these objects that show the manifestly positive nature of some alternating series. We generalize the proven identities and conjecture new polynomial refinements of Andrews-Gordon and Bressoud identities, which are companions to Foda-Quano's refinements. Finally, using a variant of the Bailey lemma, we present many new infinite hierarchies of polynomial identities.

A MacMahon Analysis View of Cylindric Partitions

TL;DR

This work applies MacMahon partition analysis and the Omega operator to study cylindric partitions with two-element profiles, deriving explicit refined generating functions and recurrences for CP_{(c1,c2)}(n) and their bilateral, manifestly positive refinements P_{(c1,c2)}(n). It connects these finite-form identities to classical Rogers–Ramanujan–type theorems and their polynomial refinements (Andrews–Gordon, Bressoud, Foda–Quano) and shows how limits yield Borodin’s product formulas, while providing several explicit bilateral sums with nonnegative coefficients. The authors prove numerous cases (notably k=2,3,4) and formulate conjectures for infinite hierarchies that they partially realize via Bailey machinery, offering a program to establish broad families of polynomial identities. The results illuminate deep links between cylindric partitions, q-series, and Bailey-type techniques, and they propose concrete pathways to extend polynomial refinements and hierarchies beyond the proven cases.

Abstract

We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analysis and then solving them. We also note some q-series identities related to these objects that show the manifestly positive nature of some alternating series. We generalize the proven identities and conjecture new polynomial refinements of Andrews-Gordon and Bressoud identities, which are companions to Foda-Quano's refinements. Finally, using a variant of the Bailey lemma, we present many new infinite hierarchies of polynomial identities.

Paper Structure

This paper contains 20 sections, 41 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Cylindric Partitions with profile $(c_1,c_2)$ such that $c_1+c_2=2$
  4. Profile $(1,1)$
  5. Profile $(2,0)$
  6. $k=2$ cases of \ref{['eq:LiUncu_FinBressoud']}
  7. Cylindric Partitions with profile $(c_1,c_2)$ such that $c_1+c_2=3$
  8. Profile $(2,1)$
  9. Profile $(3,0)$
  10. A polynomial identity due to George Andrews
  11. Cylindric Partitions with profile $(c_1,c_2)$ such that $c_1+c_2=4$
  12. Profile $(4,0)$
  13. Profile $(3,1)$
  14. Profile $(2,2)$
  15. Cylindric Partitions with profile $(c_1,c_2)$ such that $c_1+c_2=5$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $n\geq0$ and $a\geq b \geq 0$ be integers and let $CP_{(a,b)}(n)$ be the generating function for the number of cylindric partitions with profile $(a,b)$ where each partition in the cylindric partition has at most $n$ parts. We always take $CP_{(a,b)}(0)=1$ as the cylindric partition of the empty where the $q$-Pochhammer symbol $(a;q)_n$ and the $q$-binomial coefficients ${a\brack b}_q = (q;q)_

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2: Foda--Quano, 1994
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5: Foda-Quano, 1994
  • Conjecture 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Theorem 2.2: Borodin, 2007
  • ...and 45 more