Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On partitions with bounded largest part and fixed integral GBG-rank modulo primes

Alexander Berkovich, Aritram Dhar

TL;DR

The paper studies partitions with bounded largest part under fixed integral GBG-rank modulo a prime $t$ by employing the Littlewood decomposition into a $t$-core and a $t$-quotient. It provides new generating function formulas for unrestricted partitions, self-conjugate partitions, and partitions with parts repeated no more than $t-1$ times, proving four main theorems that unify these cases under a single combinatorial framework. The results include explicit $q$-product expressions and extensions to GBG-rank values of the form $k\omega_t^j$, offering a direct combinatorial approach to prior $q$-series prompts and yielding new, elegant generating-function identities. The methods illuminate the structure of GBG-rank modulo primes via rim-hook removals and the $t$-core/$t$-quotient decomposition, with potential implications for further refinements and related partition statistics.

Abstract

In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo $t$ which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new elegant generating function formulas for unrestricted partitions, self-conjugate partitions, and partitions whose parts repeat a finite number of times.

On partitions with bounded largest part and fixed integral GBG-rank modulo primes

TL;DR

The paper studies partitions with bounded largest part under fixed integral GBG-rank modulo a prime by employing the Littlewood decomposition into a -core and a -quotient. It provides new generating function formulas for unrestricted partitions, self-conjugate partitions, and partitions with parts repeated no more than times, proving four main theorems that unify these cases under a single combinatorial framework. The results include explicit -product expressions and extensions to GBG-rank values of the form , offering a direct combinatorial approach to prior -series prompts and yielding new, elegant generating-function identities. The methods illuminate the structure of GBG-rank modulo primes via rim-hook removals and the -core/-quotient decomposition, with potential implications for further refinements and related partition statistics.

Abstract

In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new elegant generating function formulas for unrestricted partitions, self-conjugate partitions, and partitions whose parts repeat a finite number of times.
Paper Structure (11 sections, 5 theorems, 76 equations, 2 figures)

This paper contains 11 sections, 5 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. Background on Littlewood decomposition of partitions
  3. Proofs of Theorems \ref{['thm11']}-\ref{['thm14']}
  4. Proof of Theorem \ref{['thm11']}
  5. Proof of Theorem \ref{['thm12']}
  6. Proof of Theorem \ref{['thm13']}
  7. Proof of Theorem \ref{['thm14']}
  8. Further Observations
  9. Acknowledgments
  10. Declaration of Interest Statement
  11. Data Availability Statement

Key Result

Theorem 1.1

For any prime $t$, a non-negative integer $N$, and any integer $k$, we have where $\tilde{q} := q^t$, and $\nu\in\{0,1,2,\ldots,t-1\}$.

Figures (2)

  • Figure 1: $3$-residue diagram of the partition $\pi = (10,7,4,3)$
  • Figure 2: Generic extended $t$-residue diagram of a partition $\pi = (\lambda_1,\lambda_2,\lambda_3,\ldots)\in\mathcal{P}$ with $\lambda_1 = tN+\nu$ where $0\le\nu\le t-1$ (the regions are labeled in red).

Theorems & Definitions (9)

  • Remark 1
  • Theorem 1.1
  • Remark 2
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 3
  • Remark 4
  • Theorem 4.1