The 3-adic valuations of Stirling numbers of the first kind

2021/09/28 by Qiu, Min, Feng, Yulu, Hong, Shaofang
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.13458

Abstract

Let n and k be positive integers. The Stirling number of the first kind, denoted by s(n,k), counts the number of permutations of n elements with k disjoint cycles. Let p be a prime number and denote by v_p(n) the p-adic valuation of n. In recent years, Lengyel, Komatsu, Young, Leonetti and Sanna made some progress on v_p(s(n,k)). Let a∈\1,2\. In this paper, by using the properties of m-th Stirling number of the first kind developed previously and providing a detailed 3-adic analysis, we arrive at an explicit formula on v₃(s(a3ⁿ, k)) with 1≤ k≤ a3ⁿ. This gives an evidence to a conjecture of Hong and Qiu proposed in 2020. As a corollary, we show that v₃(s(a3ⁿ,a3ⁿ-k))=2n-1+v₃(k)-v₃(k-1) if k is odd and 3≤ k≤ a3^n-1+1. This supports a conjecture of Lengyel raised in 2015.

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