Arithmetic properties of ℓ-regular partitions

Abstract

For a given prime p, we study the properties of the p-dissection identities of Ramanujan's theta functions ψ(q) and f (−q), respectively. Then as applications, we find many infinite family of congruences modulo 2 for some ℓ-regular partition functions, especially, for ℓ = 2, 4,5,8,13,16. Moreover, based on the classical congruences for p(n) given by Ramanujan, we obtain many more congruences for some ℓ-regular partition functions.

Congruences for (3,11)-regular bipartitions modulo 11

“…The arithmetic of -regular partition functions has received a great deal of attention (see, for example, [1,2,5,10,[12][13][14][15]20,22,[24][25][26][27]30]). Recently, Xia and Yao [31] established several infinite families of congruences modulo 2 for b 9 (n).…”

Arithmetic properties of bipartitions with 3-cores

“…Recently, Cui and Gu [10] established the p-dissection formulas of f 2 2 f 1 and f 1 . They proved that for any odd prime p ≥ 5,…”

Congruences for 9-regular partitions modulo 3

“…By means of the following lemma given by the authors in [6], we derive some new congruences for b 9 (n). …”

“…Recently, arithmetic properties of -regular partition functions have received a great deal of attention (see, for example, [2,[4][5][6][7]10,12,[14][15][16]18,19]). Xia and Yao [18] obtained that for all integers n ≥ 0 and k ≥ 0, b 9 2 6k+7 n + 2 6k+6 − 1 3 ≡ 0 (mod 2).…”