Antichain Simplices
Benjamin Braun and Brian Davis
Department of Mathematics
University of Kentucky
Lexington, KY 40506-0027
USA
Abstract:
Associated with each lattice simplex Δ is a poset encoding the
additive structure of lattice points in the fundamental parallelepiped
for Δ. When this poset is an antichain, we say Δ is antichain. For
each partition λ of n, we define a lattice simplex
Δλ having one
unimodular facet, and we investigate their associated posets. We give
a number-theoretic characterization of the relations in these posets,
as well as a simplified characterization in the case where each part
of λ is relatively prime to n − 1. We use these characterizations
to experimentally study Δλ
for all partitions of n with n ≤
73. Further, we experimentally study the prevalence of the antichain
property among simplices with a restricted type of Hermite normal form,
suggesting that the antichain property is common among simplices with
this restriction. Finally, we explain how this work relates to
Poincaré
series for the semigroup algebra associated with Δ and we prove
that this series is rational when Δ is antichain.
Full version: pdf,
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(Concerned with sequences
A002182
A004394
A040047
A323256
A323257.)
Received January 10 2019; revised version received May 16 2019; November 12 2019.
Published in Journal of Integer Sequences,
December 28 2019.
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