Geometrical Techniques for Estimating Numbers of Linear Extensions

LetPbe a two-dimensional order, and __Pany complement ofP, i.e., any partial order whose comparability graph is the complement of the comparability graph ofP. Lete(Q) denote the number of linear extensions of the partial orderQ. Sidorenko showed thate(P)e(__P) ≥ n!, for any two-dimensional partial orderP. In this note, we use results from polyhedral combinatorics, and from the geometry ofRn, to give a companion upper bound one(P)e(__P), as well as an alternative proof of the lower bound. We use these results to obtain bounds on the number of linear extensions of a random two-dimensional partial order.