Geodesic
On the cross-product conjecture for the number of linear extensions
Canadian journal of mathematics, Tome 77 (2025) no. 2, pp. 535-562

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DOI

We prove a weak version of the cross-product conjecture: $\textrm {F}(k+1,\ell ) \hskip .06cm \textrm {F}(k,\ell +1) \ge (\frac 12+\varepsilon ) \hskip .06cm \textrm {F}(k,\ell ) \hskip .06cm \textrm {F}(k+1,\ell +1)$, where $\textrm {F}(k,\ell )$ 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.
DOI : 10.4153/S0008414X24000087
Mots-clés : Poset, linear extensions, cross-product conjectures, mixed volumes, correlation inequality 
Chan, Swee Hong; Pak, Igor; Panova, Greta. On the cross-product conjecture for the number of linear extensions. Canadian journal of mathematics, Tome 77 (2025) no. 2, pp. 535-562. doi: 10.4153/S0008414X24000087
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     title = {On the cross-product conjecture for the number of linear extensions},
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     year = {2025},
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     doi = {10.4153/S0008414X24000087},
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