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arXiv · 2601.07650

On the number of antichains in $\{0,1,2\}^n$

Matthew Jenssen·Jinyoung Park·Michail Sarantis
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Abstract

We provide precise asymptotics for the number of antichains in the poset $\{0,1,2\}^n$, answering a question of Sapozhenko. Finding improved estimates for this number was also a problem suggested by Noel, Scott, and Sudakov, who obtained asymptotics for the logarithm of the number. Key ingredients for the proof include a graph-container lemma to bound the number of expanding sets in a class of irregular graphs, isoperimetric inequalities for generalizations of the Boolean lattice, and methods from statistical physics based on the cluster expansion.

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Matthew Jenssen, Jinyoung Park, Michail Sarantis. 2026-01-12. On the number of antichains in $\{0,1,2\}^n$. https://arxiv.org/abs/2601.07650

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