Permutations statistics of indexed and poset permutations

The deenitions of descents and excedances in the elements of the symmetric group S d are generalized in two diierent directions. First, descents and excedances are deened for indexed permutations, i.e. the elements of the group S n d = Z n oS d , where o is wreath product with respect to the usual action of S d by permutation of d]. It is shown, bijectively, that excedances and descents are equidistributed, and the corresponding descent polynomials, analogous to the Eulerian polynomials, are computed as the f-eulerian polynomials of simple polynomials. The descent polynomial is shown to equal the h-polynomial (essentially the h-vector) of a certain triangulation of the unit d-cube. This is proved by a bijection which exploits the fact that the h-vector of the triangulation in question can be computed via a shelling of the simplicial complex arising from the triangulation. The h-vector, in turn, is computed via the Ehrhart polynomials of dilations of the unit d-cube. The famous formula P d0 E d x d d! = sec x + tan x, where E d is the number of alternating permutations in S d , is generalized in two diierent ways, one relating to recent work of V.I. Arnold on Morse theory. The resulting formulas are then used to nd, in two special cases, a relation between the number of alternating indexed permutations and the value of the corresponding descent polynomial at-1. The deenitions of major index and inversion index are also generalized and their equidistribution is proved. Secondly, descents and excedances are generalized to all nite posets, the classical case corresponding to the poset f1; 2; : : :; dg of natural numbers in their usual ordering. Again, descents and excedances are equidistributed, which is proved bijectively. This bijection, which is not a generalization of the one described by Foata and Sch utzenberger in the classical case, has the virtue of translating descents "verbatim" into excedances. Using this, bijective proofs are given of two results concerning the chromatic polynomial of the incomparability graph of a poset P. Acknowledgements There are many people who deserve my thanks. I will mention only some of those who are directly related to my mathematical education and the writing of this thesis. My advisor, Richard Stanley, for always pointing me in the right direction and constantly suggesting new and worthwhile problems. Also for introducing me to the kind of combinatorics …