The cross-product conjecture for width two posets

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.Comment: 31 pages, 7 figures. Counterexamples to Conjecture 11.2, 11.3, and 11.4 in v2 were found, and are now included in Section 11.5 and 11.10 of v3. To appear in Trans. Amer. Math. So