A proof of a conjecture of Mao on Beck's partition statistics modulo 8

TL;DR

This paper proves a conjecture by Mao regarding specific identities involving Beck's partition statistics, confirming their congruence properties modulo 8 and 4, and advancing understanding of partition theory.

Contribution

The paper establishes a new identity linking partition statistics $NT(r,8,n)$ and $M_{ ext{omega}}(r,4,n)$, confirming Mao's conjecture and expanding the theoretical framework of partition congruences.

Findings

Proved an identity relating $NT(r,8,n)$ and $M_{\omega}(r,4,n)$

Confirmed Mao's conjecture on partition statistics

Enhanced understanding of partition congruences modulo 8 and 4

Abstract

Beck introduced two partition statistics $NT(r,m,n)$ and $M_{\omega}(r,m,n)$,which denote the total number of parts in the partition of $n$ with rank congruent to $r$ modulo $m$ and the total number of ones in the partition of $n$ with crank congruent to $r$ modulo $m$, respectively. In recent years, a number of congruences and identities on $NT(r,m,n)$ and $M_{\omega}(r,m,n)$ for some small $m $ have been established.In this paper, we prove an identity on $NT(r,8,n)$ and $M_{\omega}(r,4,n)$ which confirm a conjecture given by Mao.