Variations of Andrews-Beck type congruences

Proofs of some conjectures of Chan-Mao-Osburn on Beck's partition statistics

“…The aim of the paper is to confirm the remainder three conjectures of Chan-Mao-Osburn [6] and two conjectures due to Mao [15] on some relations involving N T (m, 5, n) and M ω (m, 5, n). Identities (1.4), (1.6) and (1.7) were first conjectured by Chan, Mao and Osburn [6] and (1.5) and (1.8) were conjectured by Mao [15]. Moreover, identity (1.6) implies (1.3).…”

“…Those conjectures on Andrews-Beck type congruences of N T (m, j, n) and M ω (m, j, n) were proved by Chern [9]. Later, Mao [14] proved the following two identities on N T (m, j, n) which were conjectured by Chan, Mao and Osburn [6]:…”

“…In a recent paper, Lin, Peng and Toh [13] considered the generalized crank defined by Fu and Tang [11] for k-colored partitions and derived a number of Andrews-Beck type congruences. Very recently, Chan, Mao and Osburn [6] proved three variations of Andrews-Beck type congruences and posed a number of conjectures on N T (m, j, n) and M ω (m, j, n). Those conjectures on Andrews-Beck type congruences of N T (m, j, n) and M ω (m, j, n) were proved by Chern [9].…”

Andrews-Beck Type Congruences for Overpartitions

“…Proof. Applying Proposition 2.2 in [5] (a generalization [1, Theorem 3]) with setting d = 1 and e → 0, we can rewrite (2.1) as follows.…”

“…where (λ) is the largest part of λ, #(λ) is the number of parts in λ, #(λ o ) is the number of odd non-overlined parts of λ, and χ(λ) = 1 if the largest part of λ is odd and non-overlined and χ(λ) = 0 otherwise. Let N T (b, k, n) denote the total number of parts in the overpartitions of n with Drank congruent to b modulo k and N T 2(b, k, n) denote the total number of parts in the overpartitions of n with M 2 -rank congruent to b modulo k. Then the following congruences are proved by Chan-Mao-Osburn [5]: for all n ∈ N,…”