Divisibility by 2 of partial Stirling numbers

2011/09/22 by Donald M. Davis, Davis, Donald M.
Mathematics · #05A99 #11B73 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A99 #msc:11B73

paper · pdf · doi:10.48550/arxiv.1109.4879

33 pages

arxiv created 2011/09/22 · arxiv updated 2011/09/23

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.

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