Divisibility by 2 of partial Stirling numbers

TL;DR

This paper investigates the 2-adic divisibility properties of partial Stirling numbers, providing new general results and interpretations within 2-adic analysis relevant to algebraic topology.

Contribution

It introduces new formulas and polynomial families to analyze 2-adic valuations of partial Stirling numbers and interprets these numbers as functions on 2-adic integers.

Findings

nu(T_n(k)) follows a specific pattern mod 2^t

k0 is a 2-adic integer related to zeros of the function

New polynomial families aid in valuation analysis

Abstract

The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose 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 - k0) + c0, where k0 is a 2-adic integer and c0 a positive integer. Our analysis involves several new general results for nu(sum (n choose 2i+1) i^j), the proofs of which involve a new family of polynomials. Following Clarke, we interpret T_n as a function on the 2-adic integers, and the 2-adic integers k0 described above as the zeros of these functions.