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Asymptotic Formula for $(t+1)$-Regular Partitions

Jayanta Barman, Kamalakshya Mahatab

Abstract

A partition is $t$-regular if none of its parts is divisible by $t$. Let $p(N,t)$ be the number of $(t+1)$-regular partitions of a positive integer $N$. In 1971, Hagis proved an asymptotic formula for $p(N,t)$ using the circle method, when $t$ fixed. In this article, we use the saddle point method and extend the result of Hagis in different ranges of $t$, obtaining explicit bounds. We also discuss an application of our result to estimate zeros in the character table of the symmetric group.

Asymptotic Formula for $(t+1)$-Regular Partitions

Abstract

A partition is -regular if none of its parts is divisible by . Let be the number of -regular partitions of a positive integer . In 1971, Hagis proved an asymptotic formula for using the circle method, when fixed. In this article, we use the saddle point method and extend the result of Hagis in different ranges of , obtaining explicit bounds. We also discuss an application of our result to estimate zeros in the character table of the symmetric group.
Paper Structure (5 sections, 16 theorems, 169 equations)

This paper contains 5 sections, 16 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Applications of Theorem 1.2
  3. Acknowledgments
  4. Preliminaries
  5. Proofs of Main Results

Key Result

Theorem 1.1

Let $N$ be a positive integer and $t \ge 4$. Then

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: hagis1971partitions
  • Theorem 1.4
  • Corollary 1.5: mcspirit2023zeros
  • Corollary 1.6
  • proof
  • Proposition 3.1
  • Lemma 3.2: tyler2024asymptotics
  • Lemma 3.3: barman2025lower
  • ...and 25 more