Dumont's Statistic on Words

Abstract

We define Dumont's statistic on the symmetric group $S_n$ to be the function dmc: $S_n \rightarrow {\bf N}$ which maps a permutation $\sigma$ to the number of distinct nonzero letters in code$( \sigma )$. Dumont showed that this statistic is Eulerian. Naturally extending Dumont's statistic to the rearrangement classes of arbitrary words, we create a generalized statistic which is again Eulerian. As a consequence, we show 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 $J(P)$. This strengthens for products of chains a result of Stanley concerning the flag $h$-vectors of Cohen-Macaulay complexes. We conjecture that the result holds for all finite distributive lattices.