Congruences modulo arbitrary powers of 5 and 7 for Andrews and Paule's partition diamonds with (n+1) copies of n

2025/02/14 by Du, Julia Q. D., Yao, Olivia X. M.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.00004

Abstract

Recently, Andrews and Paule introduced a partition function PDN1(N) which denotes the number of partition diamonds with (n+1) copies of n where summing the parts at the links gives N. They also presented the generating function for PDN1(n) and proved several congruences modulo 5,7,25,49 for PDN1(n). At the end of their paper, Andrews and Paule asked for determining infinite families of congruences similar to Ramanujan's classical p(5^kn +d_k) ≡ 0 \pmod 5^k, where 24d_k≡ 1 \pmod 5^k and k≥ 1. In this paper, we give an answer of Andrews and Paule's open problem by proving three congruences modulo arbitrary powers of 5 for PDN1(n). In addition, we prove two congruences modulo arbitrary powers of 7 for PDN1(n), which are analogous to Watson's congruences for p(n).

Related

• The k-elongated plane partition function modulo small powers of 5
• Congruences modulo powers of 7 for k-elongated plane partitions
• Congruences for k-elongated plane partition diamonds
• Congruences modulo 7 and 11 for certain two restricted partition functions
• Congruences for Andrews' spt-function modulo powers of 5, 7 and 13