On the p-adic properties of Stirling numbers of the first kind

Abstract

Let n, k and a be positive integers. The Stirling numbers of the first kind, denoted by s(n, k), count the number of permutations of n elements with k disjoint cycles. Let p be a prime. Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the p-adic valuations of s(n, k). In this paper, by using Washington’s congruence on the generalized harmonic number and the n-th Bernoulli number B_n and the properties of m-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound on v_p(s(ap, k)) witha and k being integers such that $1\le a\le p-1$ and $1\le k\le ap$. This infers that for any regular prime $p\ge 7$ and for arbitrary integers a and k with $5\le a\le p-1$ and $a-2\le k\le ap-1$, one has $v_{p}(H(ap-1,k)) < -\frac{\log{(ap-1)}}{2\log p}$ with $H(ap-1, k)$ being the k-th elementary symmetric function of $1, \frac{1}{2}, \ldots , \frac{1}{ap-1}$ . This gives a partial support to a conjecture of Leonetti and Sanna. We also present results on $v_p(s(ap^{n},ap^{n}-k))$ from which one can derive that under certain condition, for any prime $p\ge 5$, any odd number $k\ge 3$ and any sufficiently large integer n, if $(a,p)=1$, then $v_p(s(ap^{n+1},ap^{n+1}-k))=v_p(s(ap^{n},ap^{n}-k))+2$. It confirms partially Lengyel’s conjecture.