一些特殊第二类Stirling数的p-adic赋值

On the p-adic valuations of some special Stirling numbers of the second kind

设k和n为非负整数.第二类Stirling数表示将n个元素划分为恰好k个非空集合的个数,记为S(n,k).对任意给定的素数p和正整数n,存在惟一的整数a和m≥0使得n=apm,其中(a,p)=1(a与p互素).称m为n的p-adic赋值,并记vp(n)=m.第二类Stirling数的p-adic赋值是数论和代数拓扑领域的重要问题.本文研究了一些特殊第二类Stirling数S(p^(n),2^(t)p)的p-adic赋值,其中p为奇素数,t和n为正整数.本文证明当n≥2,2≤2^(t)<p时vp(S(p^(n),2^(t)p))≥n+2-2^(t),推广了Zhao和Qiu最近的结果.

Let k and n be nonnegative integers.The Stirling number of the second kind is defined as the number of ways to partition a set of n elements into exactly k non-empty subsets,denoted by S(n,k).Given a prime p and a positive integer n,there exist unique integers a and m≥0 with(a,p)=1 such that n=apm.The number m is called p-adic valuation of n,denoted by vp(n)=m.The p-adic valuations of Stirling numbers of the second kind play a role in number theory and algebraic topology.In this paper,we study the p-adic valuations of the Stirling numbers of the second kind with the special form S(p^(n),2^(t)p),where p is an odd prime,t and n are positive integers.We show that if n≥2 and 2≤2^(t)<p,then vp(S(p^(n),2^(t)p))≥n+2-2^(t).This extends the results obtained by Zhao and Qiu recently.