The 2-adic valuations of Stirling numbers of the first kind

Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ 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 $s(2^n, k)$. We also prove that $v_2(s(2^n+1,k+1))=v_2(s(2^n,k))$ holds for all integers $k$ between 1 and $2^n$. As a corollary, we show that $v_2(s(2^n,2^n-k))=2n-2-v_2(k-1)$ if $k$ is odd and $2\le k\le 2^{n-1}+1$. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if $k\le 2^n$, then $v_2(s(2^n,k)) \le v_2(s(2^n,1))$ and $v_2(H(2^n,k))\leq -n$, 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.