Ramanujan-Type Congruences for Several Partition Functions

Abstract

Inspired by the recent work of Baruah and Ojah, we investigate several partition functions and establish some interesting Ramanujan-type congruences.

Rank and Crank Analogs for some Colored Partitions

“…About further research of arithmetic properties of cubic partitions, overcubic partitions and other colored partitions, some interesting results can be found in [18,21,23,24]. The first author [25] of the present paper generalized Hammond-Lewis birank and gave combinatorial interpretations for some colored partitions.…”

“…where γ(n) is defined by ∞ n=0 γ(n)q n = (−q;q 2 ) 4 ∞ (q 2 ;q 2 )∞ . (see [23]). It can also prove the identity γ(7n + 2) ≡ 0 (mod 7).…”

“…where α(n) is defined by ∞ n=0 α(n)q n = 1 (q;q) 2 ∞ (q 2 ;q 2 )∞ . (see [23]). Suppose is primitive 7th root of unity.…”

“…In this section, we define statistics that divide the relevant partitions into equinumerous classes and provide the combinatorial interpretation according to [17] for colored partitions congruences modulo 7 given in Toh [21], Zhang and Wang [23].…”