Dumont’s statistic on words. (English) Zbl 0982.06001

Skandera, Mark

Electron. J. Comb. 8, No. 1, Research paper R11, 19 p. (2001).

It is well known that the number of permutations in the symmetric group \(S_n\) with \(k\) descents is equal to the Euler number \(A_{n,k}\). A similar property holds for many other natural permutation statistics \(\text{stat}: S_n\rightarrow N\). Namely, a statistic is called Eulerian if \(|\{ \pi\in S_n:\text{stat}(\pi)=k\}|=A_{n,k}\). In particular, D. Dumont introduced [Duke Math. J. 41, 305-318 (1974; Zbl 0297.05004)] a statistic which maps a permutation to the number of distinct nonzero letters in its code, and showed that it is Eulerian.
In the paper under review Dumont’s statistic is extended to rearrangement classes of arbitrary words. This generalized Dumont’s statistic occurs to be again Eulerian. As a consequence, the author shows that for each distributive lattice \(J(P)\) which is a product of chains, there is a poset \(Q\) such that the \(f\)-vector of \(Q\) is the \(h\)-vector of \(J(P)\).

Reviewer: Valeriy B.Mnukhin (Suva, Fiji Islands)

MSC:

06A07	Combinatorics of partially ordered sets
68R15	Combinatorics on words
13F55	Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
06A11	Algebraic aspects of posets

Keywords:

permutation statistic; Eulerian statistic; \(f\)-vector; \(h\)-vector; Cohen-Macaulay complex; distributive lattice