A combinatorial approach to height sequences in finite partially ordered sets

Fix an element x of a finite partially ordered set P on n elements. Then let h i ( x ) be the number of linear extensions of P in which x is in position i , counting from the bottom. The sequence { h i ( x ) : 1 ≤ i ≤ n } is the height sequence of x in P . In 1982, Stanley used the Alexandrov–Fenchel inequalities for mixed volumes to prove that this sequence is log-concave, i.e., h i ( x ) h i + 2 ( x ) ≤ h i + 1 2 ( x ) for 1 ≤ i ≤ n − 2 . However, Stanley’s elegant proof does not seem to shed any light on the error term when the inequality is not tight; as a result, researchers have been unable to answer some challenging questions involving height sequences in posets. In this paper, we provide a purely combinatorial proof of two important special cases of Stanley’s theorem by applying Daykin’s inequality to an appropriately defined distributive lattice. As an end result, we prove a somewhat stronger result, one for which it may be possible to analyze the error terms when the log-concavity bound is not tight.