# Antichain Simplices

## TL;DR

This paper introduces a new combinatorial structure called antichain simplices, characterizes their associated posets, and proves the rationality of related Poincaré series, with computational and theoretical insights.

## Contribution

It provides a novel lattice simplex construction linked to partitions, characterizes their posets, and establishes the rationality of Poincaré series for antichain simplices.

## Key findings

- Characterization of relations in posets associated with $ ext{Delta}_ ext{lambda}$
- Simplified conditions when parts of $ ext{lambda}$ are coprime to $n-1$
- Proof of rationality of Poincaré series for antichain simplices

## Abstract

To each lattice simplex $\Delta$ we associate a poset encoding the additive structure of lattice points in the fundamental parallelepiped for $\Delta$. When this poset is an antichain, we say $\Delta$ is antichain. To each partition $\lambda$ of $n$, we associate a lattice simplex $\Delta_\lambda$ 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 $\lambda$ is relatively prime to $n-1$. We use these characterizations to experimentally study $\Delta_\lambda$ for all partitions of $n$ with $n\leq 73$. We also investigate the structure of these posets when $\lambda$ has only one or two distinct parts. Finally, we explain how this work relates to Poincar\'e series for the semigroup algebra associated to $\Delta$, and we prove that this series is rational when $\Delta$ is antichain.