High order congruences for M-ary partitions

2024/03/07 by Błażej Żmija, Żmija, Błażej
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2403.04495

Abstract

For a sequence M=(m_i)_i=0^∞ of integers such that m₀=1, m_i≥ 2 for i≥ 1, let p_M(n) denote the number of partitions of n into parts of the form m₀m₁⋯ m_r. In this paper we show that for every positive integer n the following congruence is true: p_M(m₁m₂⋯ m_rn-1)≡ 0 (\rm mod ∏_t=2^rM(m_t,t-1)), where M(m,r):=\fracmgcd(m,\rm lcm (1,… ,r)). Our result answers a conjecture posed by Folsom, Homma, Ryu and Tong, and is a generalisation of the congruence relations for m-ary partitions found by Andrews, Gupta, and Rødseth and Sellers.

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