Linear extensions of ranked posets, enumerated by descents. A problem of Stanley from the 1981 Banff conference on ordered sets
In reviewing this paper, where the author solves a problem of Richard P. Stanley he himself might have solved based on his own work as noted by the author, one gets the sense that the write-up and the solution contains a meta-combinatorial component in its somewhat unusual but praiseworthy presentation which includes such ``wasted space'' as a very detailed worked-out example involving ideas presented in the proof of the main result as well as a short compendium of some smaller posets and their \(h\)-vectors for the student to savor. The question addressed successfully is to find a truly combinatorial proof of Stanley's generalization of Mac Mahon's theorem; i.e., to show that when \(P\) is a (finite) ranked poset, then \(h_k= h_{M-k}\), where \(M= \max\{d(\pi)\mid\pi\in L(p)\}\), \(L(P)\) is the set of linear extensions of \(P\), \(d(\pi)\) is the number of descents of the element \(\pi\in P\), and \(h_k= |H_k|\), with \(H_k\) the set of linear extensions (permutations compatible with the order on \(P\)) with \(k\) descents. After doing so, the author proposes it is possible to do better still in providing the most purely combinatorial argument in an optimally natural form. Thus does the world turn.
scientific article; zbMATH DE number 2164941