Divisibility by 2 of partial Stirling numbers
Number Theory
2011-09-23 v1 Combinatorics
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.
Cite
@article{arxiv.1109.4879,
title = {Divisibility by 2 of partial Stirling numbers},
author = {Donald M. Davis},
journal= {arXiv preprint arXiv:1109.4879},
year = {2011}
}
Comments
33 pages