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

Abstract

Let $v_3$ denote the usual $3$-adic valuation, and let $s(n, k)$ be the unsigned Stirling number of the first kind. In this paper, for $a\in\{1,2\}$, we determine the values of $v_3(s(a3^n, k))$ for all $1\le k\le a3^n$. More precisely, for each admissible pair $(m, k)$, we obtain an explicit formula for $v_3(s(a3^n, a3^m-k))$. The proof combines properties of the $m$-th Stirling numbers of the first kind with a detailed analysis of the relevant $3$-adic orders. As a consequence, we prove the case $p=3$ of a conjecture of Hong and Qiu proposed in 2020. We also derive formulas near the diagonal, comparison results for the adjacent orders $a3^n$ and $a3^n+1$, sharp upper bounds for the families $v_3(s(3^n, k))$ and $v_3(s(2\cdot3^n, k))$, and partial confirmations of conjectures of Lengyel and of Leonetti and Sanna.