Not so new congruences for Stirling numbers of the first kind, with an application to Chern classes

TL;DR

This paper derives simple binomial coefficient-based formulas for Stirling numbers of the first kind modulo prime powers and applies these results to compute Chern classes of permutation representations of cyclic groups.

Contribution

It introduces new explicit formulas for Stirling numbers modulo prime powers and applies them to algebraic topology problems involving Chern classes.

Findings

Formulas for c(n,k) mod p^{v_p(n)} involving binomials

Computed Chern classes of permutation representations of cyclic groups

Simplified expressions for Stirling numbers in modular arithmetic

Abstract

In this paper we give simple expressions, involving binomials coefficients, for the value of $c(n,k)$ modulo $p^{v_p(n)}$, when $v_p(n) > 0$. Here $c(n,k)$ denotes a Stirling number of the first kind, and $v_p(n)$ is the highest power of $p$ dividing $n$. As an application, we compute the Chern classes of permutation representations of cyclic groups.