Congruences for ℓ-regular partition functions modulo 3

Abstract

Let b (n) denote the number of -regular partitions of n. Recently Andrews, Hirschhorn, and Sellers proved that b 4 (n) satisfies two infinite families of congruences modulo 3, and Webb established an analogous result for b 13 (n). In this paper we prove similar families of congruences for b (n) for other values of .

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).…”

“…(1.8)Conjecture 1.2For any n ≥ 0,B 3,7 (An + B) ≡ 0 (mod 2), (1.9) B 3,7 (Cn + D) ≡ 0 (mod 3), (1.10) B 3,7 (En + F) ≡ 0 (mod 9),(1.11) where (A, B) ∈ {(14, 4), (14, 10), (16, 1), (28, 6), (32, 21)}, (C, D) = (4, 3), (E, F) ∈ {(7, 3),(7,4),(14,13),(21,6),(21,20),(25,3),(25,13),(25,18),(25,23)}.…”

Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7

“…Lin and Wang [11] showed that 9-regular partitions and 3-cores satisfy the same congruences modulo 3 and further generalized Keith's conjecture and derived a stronger result. Furcy and Penniston [6] obtained congruences for b (n) modulo 3 with = 4, 7, 13, 19, 25, 34, 37, 43, 49 by using the theory of modular forms.…”

Partition identities arising from Ramanujan’s formulas for multipliers

“…For example, Hirschhorn and Sellers [21], Furcy and Penniston [17], Webb [34] and Lovejoy and Penniston [29]. Recently, Cui and Gu [16], Keith [22], and Baruah and Das [5] found several arithmetic properties and infinite families of congruences for some k-regular partitions.…”