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

Abstract

Let b (n) be the number of -regular partitions of n. We show that the generating functions of b (n) with = 3, 5, 6, 7 and 10 are congruent to the products of two items of Ramanujan's theta functions ψ(q), f (−q) and (q; q) 3 ∞ modulo 3, 5 and 7. So we can express these generating functions as double summations in q. Based on the properties of binary quadratic forms, we obtain vanishing properties of the coefficients of these series. This leads to several infinite families of congruences for b (n) modulo 3, 5 and 7.

Distribution of certain $\ell$-regular partitions and triangular numbers

“…If ℓ is a positive integer, then a partition is called an ℓ-regular partition if there is no part divisible by ℓ. Many mathematicians have studied this partition function and proved several interesting arithmetic and combinatorial properties; see [6,11,22]. Furthermore, various other types of partition functions are studied in the literature by imposing certain restrictions on the parts of an ℓ-regular partitions of n. For example, Corteel and Lovejoy [5] introduced the notion of overpartition as a partition of n in which the first occurrence of a number may be overlined.…”

Ramanujan-type congruences for ℓ-regular partitions modulo 3, 5, 11 and 13

“…In this section, we give the proof of Theorem 1.2 by using the following Vanishing Property given by Hou et al [6].…”

“…Furcy and Penniston [5] obtained congruences for b ℓ (n) modulo 3 with ℓ ∈ {4, 7, 13, 19, 25, 34, 37, 43, 49} by using the theory of modular forms. Hou et al [6] proved several infinite families of congruences for b ℓ (n) modulo 3, 5 and 7 by applying binary quadratic form approach.…”

“…The objective of this paper is to derive several infinite families of congruences for ℓ-regular partition modulo 3, 5, 11 and 13. Our approaches are based on the theory of Hecke eigenforms and binary quadratic forms due to [6]. The main results of this paper are stated as follows.…”

Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

“…For example, see [AB16,CW14,DP09,Lin15,Web11]. For more studies on b (n), we refer to Hou et al [HSZ15] and the references cited there. Now, let b ;3 (n) denote the number of -regular partitions of n where a part can appear in 3 colours.…”