A combinatorial approach to height sequences in finite partially ordered sets

Fix an element xx of a finite partially ordered set PP on nn elements. Then let hi(x)hi(x) be the number of linear extensions of PP in which xx is in position ii, counting from the bottom. The sequence {hi(x):1≤i≤n}{hi(x):1≤i≤n} is the height sequence   of xx in PP. In 1982, Stanley used the Alexandrov–Fenchel inequalities for mixed volumes to prove that this sequence is log-concave, i.e., hi(x)hi+2(x)≤hi+12(x) for 1≤i≤n−21≤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.