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arXiv · 2104.09009

The cross-product conjecture for width two posets

Swee Hong Chan·Igor Pak·Greta Panova
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Abstract

The cross--product conjecture (CPC) of Brightwell, Felsner and Trotter (1995) is a two-parameter quadratic inequality for the number of linear extensions of a poset $P= (X, \prec)$ with given value differences on three distinct elements in $X$. We give two different proofs of this inequality for posets of width two. The first proof is algebraic and generalizes CPC to a four-parameter family. The second proof is combinatorial and extends CPC to a $q$-analogue. Further applications include relationships between CPC and other poset inequalities, including a new $q$-analogue of the Kahn--Saks inequality.

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Swee Hong Chan, Igor Pak, Greta Panova. 2021-04-19. The cross-product conjecture for width two posets. https://doi.org/10.1090/tran/8679

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