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Abstract
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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.
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Keywords
lattices of finite index, finite abelian $p$-groups, Smith
normal form, Hermite normal form, p-binomial coefficient
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Mathematical Subject Classification
Primary: 05A15, 20K01
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Milestones
Received: 6 April 2020
Revised: 31 August 2020
Accepted: 17 September 2020
Published: 16 January 2021
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