2018/12/11 by Qiu, Min, Hong, Shaofang
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1812.04539
Let n and k be positive integers. We denote by v2(n) the 2-adic valuation of n. The Stirling numbers of the first kind, denoted by s(n,k), counts the number of permutations of n elements with k disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the p-adic valuations of s(n,k). In this paper, by introducing the concept of m-th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of s(2n, k). We also prove that v2(s(2n+1,k+1))=v2(s(2n,k)) holds for all integers k between 1 and 2n. As a corollary, we show that v2(s(2n,2n-k))=2n-2-v2(k-1) if k is odd and 2≤ k≤ 2n-1+1. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if k≤ 2n, then v2(s(2n,k)) ≤ v2(s(2n,1)) and v2(H(2n,k))≤ -n, where H(n,k) stands for the k-th elementary symmetric functions of 1,1/2,...,1/n. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.