To each lattice simplex ∆ we associate a poset encoding the additive structure of lattice points in the fundamental parallelepiped for ∆. When this poset is an antichain, we say ∆ is antichain. To each partition λ of n, we associate a lattice simplex ∆ λ 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 ∆ λ 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. We also investigate the structure of these posets when λ has only one or two distinct parts. Finally, we explain how this work relates to Poincaré series for the semigroup algebra associated to ∆, and we prove that this series is rational when ∆ is antichain.

We associate to a finite digraph D a lattice polytope P D whose vertices are the rows of the Laplacian matrix of D. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of P D equals the complexity of D, and P D contains the origin in its relative interior if and only if D is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, h * -polynomial, and integer decomposition property of P D in these cases. We extend Braun and Meyer's study of cycles by considering cycle digraphs. In this setting we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.

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