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

2024/10/11 by Chen, Dandan, Chen, Rong, Yin, Siyu
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.08450

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_ω(r,11,n)-M_ω(11-r,11,n), where r=1,2,3,4,5. In particular, we discover a beautiful identity involving M_ω(r,11,11n+6).

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