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2021 A combinatorial identity for the $p$-binomial coefficient based on abelian groups
Chudamani Pranesachar Anil Kumar
Mosc. J. Comb. Number Theory 10(1): 13-24 (2021). DOI: 10.2140/moscow.2021.10.13

Abstract

For nonnegative integers kn, we prove a combinatorial identity for the p-binomial coefficient [b]nkp 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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Chudamani Pranesachar Anil Kumar. "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

Information

Received: 6 April 2020; Revised: 31 August 2020; Accepted: 17 September 2020; Published: 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.2140/moscow.2021.10.13

Subjects:
Primary: 05A15, 20K01

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.10 • No. 1 • 2021
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