Some results on <em>p</em>-adic valuations of Stirling numbers of the second kind

Abstract

Let $n$ and $k$ be nonnegative integers. The Stirling number of the second kind, denoted by $S(n, k)$, is defined as the number of ways to partition a set of $n$ elements into exactly $k$ nonempty subsets and we have $$ S(n, k)=\frac{1}{k!}\sum_{i=0}^{k}(-1)^i\binom{k}{i}(k-i)^n. $$ Let $p$ be a prime and $v_p(n)$ stand for the $p$-adic valuation of $n$, i.e., $v_p(n)$ is the biggest nonnegative integer $r$ with $p^r$ dividing $n$. Divisibility properties of Stirling numbers of the second kind have been studied from a number of different perspectives. In this paper, we present a formula to calculate the exact value of $p$-adic valuation of $S(n, n-k)$, where $n\ge k+1$ and $1\le k\le 7$. From this, for any odd prime $p$, we prove that $v_p((n-k)!S(n, n-k))< n$ if $n\ge k+1$ and $0\le k\le 7$. It confirms partially Clarke's conjecture proposed in 1995. We also give some results on $v_p(S(ap^n, ap^n-k))$, where $a$ and $n$ are positive integers with $(a, p)=1$ and $1\le k\le 7$.

Keywords

• stirling number of the second kind
• <i>p</i>-adic valuation
• stirling-like numbers
• <i>r</i>-associated stirling number of the second kind