A Combinatorial Identity for the p-Binomial Coefficient Based on Abelian Groups

For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for $r\in \mathbb{N}\cup\{0\},s\in \mathbb{N}$ and $p$ a prime, we present a purely combinatorial formula for the number of subgroups of $\mathbb{Z}^s$ of finite index $p^r$ with quotient isomorphic to the finite abelian $p$-group of type $\underline{\lambda}$, which is a partition of $r$ into at most $s$ parts. This purely combinatorial formula is similar to that for the enumeration of subgroups of a certain type in a finite abelian $p$-group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise to many enumeration formulae that involve polynomials in $p$ with non-negative integer coefficients.