A combinatorial approach to height sequences in finite partially ordered sets
نویسندگان
چکیده
Fix an element x of a finite partially ordered set P on n elements. Then let hi(x) count the number of linear extensions of P in which x is in position i, starting the count from the bottom. The sequence {hi(x) : 1 ≤ i ≤ n} is called 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) ≤ h 2 i+1 (x), for all i with 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, and 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 the Fortuin, Kasteleyn and Ginibre (FKG) correlation 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.
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عنوان ژورنال:
- Discrete Mathematics
دوره 311 شماره
صفحات -
تاریخ انتشار 2011