The Number of Linear Extensions of Ranked Posets

We revisit the question of counting the number of linear extensions of the Boolean lattice, relating this to the polyhedral methods of Kahn and Kim, and of Stanley. We give simpler proofs of various known results, and give an upper bound on the number of linear extensions of an arbitrary ranked poset satisfying the LYM condition. [Note: This preprint is not intended for journal publication, as it is a record of some alternative proofs of known theorems. The one result, Theorem 2.1, that does not appear in the literature will appear in the forthcoming paper of Brightwell and Tetali [1]. Other material here is likely to find its way into the book of Brightwell and Trotter.]