Some identities on Lin-Peng-Toh's partition statistic of k-colored partitions

2023/08/11 by Lin, Yang, Ernest X. W. Xia, Xia, Ernest X. W. +2
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2308.05931

openalex publication_date 2023/08/11 · openalex created_date 2023/08/15 · openalex updated_date 2026/07/28

Abstract

Recently, Andrews proved two conjectures on a partition statistic introduced by Beck. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and To introduced a partition statistic of k-colored partitions NB_k(r,m,n) which counts the total number of parts of π⁽¹⁾ in each k-colored partition π of n with \rm crank_k(π) congruent to r modulo m and proved a number of congruences for NB_k(r,m,n). In this paper, we prove some identities on NB_k(r,m,n) which are analogous to Ramanujan's ``most beautiful identity". Moreover, those identities imply some congruences proved by Lin, Peng and Toh.

Related

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