Divisibility by 2 of Stirling numbers of the second kind and their differences

Abstract

Let $n,k,a$ and $c$ be positive integers and $b$ be a nonnegative integer. Let $\nu_2(k)$ and $s_2(k)$ be the 2-adic valuation of $k$ and the sum of binary digits of $k$, respectively. Let $S(n,k)$ be the Stirling number of the second kind. It is shown that $\nu_2(S(c2^n,b2^{n+1}+a))\geq s_2(a)-1,$ where $0<a<2^{n+1}$ and $2\nmid c$. Furthermore, one gets that $\nu_2(S(c2^{n},(c-1)2^{n}+a))=s_2(a)-1$, where $n\geq 2$, $1\leq a\leq 2^n$ and $2\nmid c$. Finally, it is proved that if $3\leq k\leq 2^n$ and $k$ is not a power of 2 minus 1, then $\nu_2(S(a2^{n},k)-S(b2^{n},k))=n+\nu_2(a-b)-\lceil\log_2k\rceil +s_2(k)+\delta(k), $ where $\delta(4)=2$, $\delta(k)=1$ if $k>4$ is a power of 2, and $\delta(k)=0$ otherwise. This confirms a conjecture of Lengyel raised in 2009 except when $k$ is a power of 2 minus 1.