THE CROSS-PRODUCT CONJECTURE FOR WIDTH TWO POSETS

The cross-product conjecture (CPC) of Brightwell, Felsner and Trotter [Order 12 (1995), pp. 327-349] is a two-parameter quadratic inequality for the number of linear extensions of a poset P = (X, (sic) )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, and the equality part of the CPC for posets of width two.