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Parity Considerations for Mex-Related Partition Functions of Andrews and Newman

Robson da Silva, James A. Sellers

Abstract

In a recent paper, Andrews and Newman extended the mex-function to integer partitions and proved many partition identities connected with these functions. In this paper, we present parity considerations of one of the families of functions they studied, namely $p_{t,t}(n)$. Among our results, we provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$.

Parity Considerations for Mex-Related Partition Functions of Andrews and Newman

Abstract

In a recent paper, Andrews and Newman extended the mex-function to integer partitions and proved many partition identities connected with these functions. In this paper, we present parity considerations of one of the families of functions they studied, namely . Among our results, we provide complete parity characterizations of and .

Paper Structure

This paper contains 4 sections, 11 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$
  3. Additional congruences
  4. Closing thoughts

Key Result

Theorem \oldthetheorem

For all $n\geq 1,$$p_{1,1}(n)$ equals the number of partitions of $n$ with non-negative crank.

Theorems & Definitions (19)

  • Theorem \oldthetheorem: A-N, Theorem 2
  • Theorem \oldthetheorem: A-N, Theorem 3
  • Lemma 1
  • proof
  • Theorem \oldthetheorem
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • ...and 9 more