Congruences involving F‐partition functions

Abstract

ABSTRACT. The primary goal of this note is to prove the congruence Ca (:3n + 2) 0 (mod 3), where a (n) denotes the number of F-partitions of n with at most :3 repetitions. Secondarily, we conjecture a new family of congruences involving c2 (n), the number of F-partitions of n with 2 colors.

On certain weighted 7-colored partitions

“…From Ramanujan's congruences for p(n) modulo powers of 5, it is easy to get c 4 (5 α n + λ α ) ≡ 0 (mod 5 α ), where λ α is the least positive reciprocal of 12 modulo 5 α . Interestingly, this family of congruences resembles Sellers' result[19] for cφ 2 (n):…”

Computational proofs of congruences for 2‐colored Frobenius partitions

“…Certainly, a proof of Conjecture 1.1 via modular forms or generating function manipulations is still desired. This was originally requested in [9], and we renew that request here, given the new computational information that is now known about this partition function and the fact that Theorem 1.2 is proven.…”

“…In this paper, we will focus our attention on one of these functions, namely cφ 2 (m), which denotes the number of generalized Frobenius partitions of m with 2 colors. In [2], Andrews gives the generating function for cφ 2 More recently, Sellers [9] conjectured the following infinite family of congruences satisfied by cφ 2 . Conjecture 1.1.…”

Congruences modulo powers of 5 for the rank parity function

“…This enables us to use the theory of modular functions to obtain congruences. Our presentations and methods are similar to those of Paule and Radu [PaRa12], who solved a difficult conjecture of Sellers [Se94] for congruences modulo powers of 5 for Andrews's two-colored generalized Frobenius partitions [An84]. In Section 2. we include the necessary background and algorithms from the theory of modular functions for proving identities.…”