The 2-adic valuations of Stirling numbers of the second kind

TL;DR

This paper studies the 2-adic valuations of Stirling numbers of the second kind, confirming two conjectures related to their divisibility properties and binary digit sums.

Contribution

It proves two conjectures by Amdeberhan, Manna, and Moll regarding 2-adic valuations of Stirling numbers of the second kind.

Findings

Established the condition for equality of valuations $v_2(S(4i, 5))$ and $v_2(S(4i+3, 5))$

Proved that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for all positive integers $n$

Confirmed conjectures on divisibility and binary digit sum relations of Stirling numbers

Abstract

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.