Abstract

For nonnegative integers $k\le n$, we prove a combinatorial identity for the $p$-binomial coefficient $\left[b\right]{\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}_p$ based on abelian $p$-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for $r\in \mathbb{N}\cup \left\{0\right\},s\in \mathbb{N}$ 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 $\underset{¯}{\lambda }$ , 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 nonnegative integer coefficients.

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