Proofs of two conjectural Andrews–Beck type congruences due to Lin, Peng and Toh

Abstract

The study of Andrews–Beck type congruences for partitions has its origin in the work by Andrews, who proved two congruences on the total number of parts in the partitions of [Formula: see text] with the Dyson rank, conjectured by George Beck. Recently, Lin, Peng and Toh proved many Andrews–Beck type congruences for [Formula: see text]-colored partitions. Moreover, they posed eight conjectural congruences. In this paper, we confirm two congruences modulo [Formula: see text] by utilizing some [Formula: see text]-series techniques and the theory of modular forms.

A proof of Mao’s conjecture on an identity of Beck’s partition statistics

“…Motivated by Andrews' work, a number of identities and congruences on N T (r, m, n) and M ω (r, m, n) and their variations have been proved; see for example [9,10,11,12,14,15,17,19,20,21,22,23,27,28]. Very recently, Mao [20] proved some identities on the total number of parts functions associated to ranks of overpartition.…”