Vol. 10, No. 1, 2021

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A combinatorial identity for the $p$-binomial coefficient based on abelian groups

Chudamani Pranesachar Anil Kumar

Vol. 10 (2021), No. 1, 13–24
Abstract

For nonnegative integers k n, we prove a combinatorial identity for the p-binomial coefficient [b]n k p based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for r {0},s and p a prime, we present a purely combinatorial formula for the number of subgroups of s of finite index pr with quotient isomorphic to the finite abelian p-group of type λ¯ , 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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Keywords
lattices of finite index, finite abelian $p$-groups, Smith normal form, Hermite normal form, p-binomial coefficient
Mathematical Subject Classification
Primary: 05A15, 20K01
Milestones
Received: 6 April 2020
Revised: 31 August 2020
Accepted: 17 September 2020
Published: 16 January 2021
Authors
Chudamani Pranesachar Anil Kumar
School of Mathematics
The Harish-Chandra Research Institute
India