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 ${\nu }_2(S(c{2}^n,b{2}^{n+1}+a))\ge s_2(a)-1,$ where $0<a<2^{n+1}$ and $2\nmid c$. Furthermore, one gets that ${\nu }_2(S(c{2}^n,(c-1)2^n+a))=s_2(a)-1$, where $n\ge 2$, $1\le a\le 2^n$ and $2\nmid c$. Finally, it is proved that if $3\le k\le 2^n$ and $k$ is not a power of 2 minus 1, then ${\nu }_2(S(a{2}^n,k)-S(b{2}^n,k))=n+{\nu }_2(a-b)-\lceil {\log }_{2}k\rceil +s_2(k)+\delta (k),$ 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.

arXiv comment23 pages. To appear in Journal of Number Theory

References in corpus (1)

• Congruence classes of 2-adic valuations of Stirling numbers of the second kind