On the total number of ones associated with cranks of partitions modulo 11

TL;DR

This paper investigates the partition statistic related to the total number of ones in partitions with a crank modulo 11, deriving new identities and dissection formulas for these statistics.

Contribution

It establishes the 11-dissection of generating functions for specific crank-related partition statistics and uncovers a new identity involving these statistics at 11n+6.

Findings

11-dissection of generating functions derived

New identity involving M_w(r,11,11n+6) discovered

Enhanced understanding of partition statistics modulo 11

Abstract

In 2021, Andrews mentioned that George Beck introduced partition statistics $M_w(r,m,n)$, which denote the total number of ones in the partition of $n$ with crank congruent to $r$ modulo $m$. Recently, a number of congruences and identities involving $M_w(r,m,n)$ for some small $m$ have been developed. We establish the 11-dissection of the generating functions for $M_{\omega}(r,11,n)-M_{\omega}(11-r,11,n)$, where $r=1,2,3,4,5$. In particular, we discover a beautiful identity involving $M_{\omega}(r,11,11n+6)$.