Vol. 10, No. 1, 2021

Download this article
Download this article For screen
For printing
Recent Issues
Volume 10, Issue 1
Volume 9, Issue 4
Volume 9, Issue 3
Volume 9, Issue 2
Volume 9, Issue 1
Volume 8, Issue 4
Volume 8, Issue 3
Volume 8, Issue 2
Volume 8, Issue 1
The Journal
About the Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
founded and published with the
scientific support and advice of the
Moscow Institute of
Physics and Technology
ISSN (electronic): 2640-7361
ISSN (print): 2220-5438
Previously Published
Author Index
To Appear
Other MSP Journals
A combinatorial identity for the $p$-binomial coefficient based on abelian groups

Chudamani Pranesachar Anil Kumar

Vol. 10 (2021), No. 1, 13–24

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.

lattices of finite index, finite abelian $p$-groups, Smith normal form, Hermite normal form, p-binomial coefficient
Mathematical Subject Classification
Primary: 05A15, 20K01
Received: 6 April 2020
Revised: 31 August 2020
Accepted: 17 September 2020
Published: 16 January 2021
Chudamani Pranesachar Anil Kumar
School of Mathematics
The Harish-Chandra Research Institute