On the cross-product conjecture for the number of linear extensions

TL;DR

This paper establishes a weak inequality related to the cross-product conjecture for counting linear extensions in posets, using geometric and combinatorial methods, and disproves a generalized version of the conjecture.

Contribution

It proves a weak form of the cross-product conjecture for linear extensions and disproves its generalized version, introducing geometric inequalities and combinatorics of words.

Findings

Proved a weak inequality for linear extensions involving fixed elements.

Disproved the generalized cross-product conjecture.

Used geometric inequalities and combinatorics of words in proofs.

Abstract

We prove a weak version of the cross--product conjecture: ${F}(k+1,\ell) {F}(k,\ell+1) \geq (\frac12+\varepsilon) {F}(k,\ell) {F}(k+1,\ell+1)$, where ${F}(k,\ell)$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are $k$ and $\ell$ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the {generalized cross--product conjecture}. The proofs use geometric inequalities for mixed volumes and combinatorics of words.