The bounds for the number of linear extensions via chain and antichain coverings

TL;DR

This paper establishes bounds on the number of linear extensions of a finite poset using chain and antichain coverings, providing new inequalities that relate these bounds to the structure of the poset.

Contribution

The paper introduces novel bounds for linear extensions based on chain and antichain coverings, extending previous combinatorial inequalities.

Findings

Lower bound on linear extensions using antichain coverage

Upper bound on linear extensions using chain coverage

Corollary relating partitioning into antichains to the number of linear extensions

Abstract

Let $(\mathcal{P},\leqslant)$ be a finite poset. Define the numbers $a_1,a_2,\ldots$ (respectively, $c_1,c_2,\ldots$) so that $a_1+\ldots+a_k$ (respectively, $c_1+\ldots+c_k$) is the maximal number of elements of $\mathcal{P}$ which may be covered by $k$ antichains (respectively, $k$ chains.) Then the number $e(\mathcal{P})$ of linear extensions of poset $\mathcal{P}$ is not less than $\prod a_i!$ and not more than $n!/\prod c_i!$. A corollary: if $\mathcal{P}$ is partitioned onto disjoint antichains of size $b_1,b_2, \ldots$, then $e(\mathcal{P})\geqslant \prod b_i!$.