Infinite Families of Divisibility Properties Modulo 4 for Non-Squashing Partitions into Distinct Parts

Abstract

In 2005, Sloane and Sellers defined a function b(n) which denotes the number of nonsquashing partitions of n into distinct parts. In their 2005 paper, Sloane and Sellers also proved various congruence properties modulo 2 satisfied by b(n). In this note, we extend their results by proving two infinite families of congruence properties modulo 4 for b(n). In particular, we prove that for all k ≥ 3 and all n ≥ 0, b(2 2k+1 n + 2 2k−2 ) ≡ 0 (mod 4) and b(2 2k+1 n + 3 · 2 2k−2 + 1) ≡ 0 (mod 4).

PED and POD partitions: Combinatorial proofs of recurrence relations

“…Much work has been done in studying arithmetic properties of PED and POD partitions. Among the may articles on the subject see, for example, [8,10,13,15,23,25,29] for congruences for PED partitions and [16,18,22,27,31] for congruences for POD partitions. Hence, this would be a natural topic of further study.…”

On the number of overpartitions into odd parts

“…For more details on overpartitions, one can refer [5], [8], [9] and [12]. Define p o (n) to be the number of overpartitions of n into odd parts.…”

Propriedades aritméticas e combinatórias de funções que contam partições

“…Andrews, Hirschhorn e Sellers [8], Chen [20], Cui e Gu [26], Hirschhorn e Sellers [31] e Xia [54] Demonstração. Temos que ped♣n 12p, 3, λ 3 ➙ 4p 1q ✏ ped♣n, 3q.…”

“…Partições em que as partes pares são distintas (ped♣nq) têm sido amplamente estudadas nos últimos anos (veja [4,8,20,26,31,54]). Em 2017, Merca [41] obteve uma relação de recorrência linear eficiente e estabeleceu conexões entre ped♣nq e sobrepartições.…”