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arXiv · 1209.6284

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

Jianrong Zhao·Shaofang Hong·Wei Zhao
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Abstract

Let $n,k,a$ and $c$ be positive integers and $b$ be a nonnegative integer. Let $ν_2(k)$ 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 $ν_2(S(c2^n,b2^{n+1}+a))\geq s_2(a)-1,$ where $0 4$ is a power of 2, and $δ(k)=0$ otherwise. This confirms a conjecture of Lengyel raised in 2009 except when $k$ is a power of 2 minus 1.

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Jianrong Zhao, Shaofang Hong, Wei Zhao. 2014-02-25. Divisibility by 2 of Stirling numbers of the second kind and their differences. https://arxiv.org/abs/1209.6284

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