Geometrical techniques for estimating numbers of linear extensions
Let P be a two-dimensional order, and P̄ any complement of P, i.e., any partial order whose comparability graph is the complement of the comparability graph of P. Let e(Q) denote the number of linear extensions of the partial order Q. Sidorenko  showed that e(P)e(P̄) ≥ n!, for any two-dimensional partial order P. In this note, we use results from polyhedral combinatorics, and from the geometry of ℝn, to give a companion upper bound on e(P)e(P̄), 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. © 1999 Academic Press.
European Journal of Combinatorics
Bollobás, B., Brightwell, G., & Sidorenko, A. (1999). Geometrical techniques for estimating numbers of linear extensions. European Journal of Combinatorics, 20 (5), 329-335. https://doi.org/10.1006/eujc.1999.0299