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Parity of parts and excludant statistics in partitions

Gargi Mukherjee

Abstract

In this paper, we study restricted excludant statistics depending on its parity in partitions where parts with same parity are distinct. Using $q$-series transformations, we show that generating functions of these partition statistics are related to the quantum modular forms $\s(q)$, its companion $σ^*(q)$ introduced by Ramanujan, and $v_2(q)$, a Nahm-type sum, introduced by Andrews. Utilizing Tauberian method, we obtain asymptotics of such sequences.

Parity of parts and excludant statistics in partitions

Abstract

In this paper, we study restricted excludant statistics depending on its parity in partitions where parts with same parity are distinct. Using -series transformations, we show that generating functions of these partition statistics are related to the quantum modular forms , its companion introduced by Ramanujan, and , a Nahm-type sum, introduced by Andrews. Utilizing Tauberian method, we obtain asymptotics of such sequences.
Paper Structure (7 sections, 17 theorems, 100 equations, 1 table)

This paper contains 7 sections, 17 theorems, 100 equations, 1 table.

Table of Contents

  1. Introduction and statement of main results
  2. Preliminary lemmas
  3. Proof of Theorems \ref{['thm_aod']}--\ref{['asymp2']}
  4. Proof of Theorems \ref{['thm_sigma']}--\ref{['thm_meex-4e']}
  5. Proof of Theorem \ref{['thm_abarod']} and Theorem \ref{['thm_sigmabar']}
  6. Proof of Theorem \ref{['evthm1']} and Theorem \ref{['evthm2']}
  7. Concluding remarks

Key Result

Theorem 1.1

We have and

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 7 more