A combinatorial approach to height sequences in finite partially ordered sets

12 years 11 months ago

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Fix an element x of a ﬁnite partially ordered set P on n elements. Then let hi(x) be the number of linear extensions of P in which x is in position i, counting from the bottom. The sequence {hi(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., hi(x)hi+2(x) ≤ h2 i+1(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 b...

Csaba Biró, William T. Trotter

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