Divisibility by 2 of partial Stirling numbers

Abstract

The partial Stirling numbers $T_n(k)$ used here are defined as $\sum_{i\text{ odd}}\binom{n}{i}i^k$. Their 2-exponents $\nu(T_n(k))$ are important in algebraic topology. We provide many specific results, applying to all values of $n$, stating that, for all $k$ in a certain congruence class mod $2^t$, $\nu(T_n(k))=\nu(k-k_0)+c_0$, where $k_0$ is a 2-adic integer and $c_0$ a positive integer. Our analysis involves several new general results for $\nu(\sum\binom{n}{2i+1}i^j)$, the proofs of which involve a new family of polynomials. Following Clarke [3], we interpret $T_n$ as a function on the 2-adic integers, and the 2-adic integers $k_0$ described above as the zeros of these functions.