Journal of Integer Sequences, Vol. 23 (2020), Article 20.1.1

Antichain Simplices

Benjamin Braun and Brian Davis
Department of Mathematics
University of Kentucky
Lexington, KY 40506-0027


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.

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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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