×

zbMATH — the first resource for mathematics

On the divisibility by 2 of the Stirling numbers of the second kind. (English) Zbl 0808.11017
Summary: We characterize the divisibility by 2 of the Stirling numbers of the second kind, \(S(n,k)\), where \(n\) is a sufficiently high power of 2. Let \(\nu_ 2(r)\) denote the highest power of 2 that divides \(r\). We show that there exists a function \(L(k)\) such that, for all \(n\geq L(k)\), \(\nu_ 2 (k! S(2^ n, k))= k-1\) hold, independently from \(n\). (The independence follows from the periodicity of the Stirling numbers modulo any prime power.) For \(k\geq 5\), the function \(L(k)\) can be chosen so that \(L(k)\leq k-2\). We determine \(\nu_ 2 (k! S(2^ n+u, k))\) for \(k>u\geq 1\), in particular for \(u=1\), 2, 3, and 4. We show how to calculate it for negative values, in particular for \(u=-1\). The characterization is generalized for \(\nu_ 2 (k! S(c\cdot 2^ n+ u,k))\), where \(c>0\) denotes an arbitrary odd integer.

MSC:
11B73 Bell and Stirling numbers
11B50 Sequences (mod \(m\))
PDF BibTeX XML Cite