On the number of antichains in $\{0,1,2\}^n$
Combinatorics
2026-01-13 v1
Abstract
We provide precise asymptotics for the number of antichains in the poset , 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.
Keywords
Cite
@article{arxiv.2601.07650,
title = {On the number of antichains in $\{0,1,2\}^n$},
author = {Matthew Jenssen and Jinyoung Park and Michail Sarantis},
journal= {arXiv preprint arXiv:2601.07650},
year = {2026}
}