For nonnegative integers , we prove a combinatorial identity for the -binomial coefficient based on abelian -groups. A purely combinatorial proof of this identity is not known. While proving this identity, for and a prime, we present a purely combinatorial formula for the number of subgroups of of finite index with quotient isomorphic to the finite abelian -group of type , which is a partition of into at most parts. This purely combinatorial formula is similar to that for the enumeration of subgroups of a certain type in a finite abelian -group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise to many enumeration formulae that involve polynomials in with nonnegative integer coefficients.
"A combinatorial identity for the $p$-binomial coefficient based on abelian groups." Mosc. J. Comb. Number Theory 10 (1) 13 - 24, 2021. https://doi.org/10.2140/moscow.2021.10.13