The cross-product conjecture for width two posets
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, ) 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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