Dumont's Statistic on Words

We define Dumont’s statistic on the symmetric group Sn to be the function dmc: Sn → N which maps a permutation σ to the number of distinct nonzero letters in code(σ). 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.