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.