Permutation Statistics of Indexed Permutations

The deenitions of descent, excedance, major index, inversion index and Denert's statistic for the elements of the symmetric group S d are generalized It is shown, bijectively, that excedances and descents are equidistributed, and the corresponding descent polynomial, analogous to the Eulerian polynomial , is computed as the f {eulerian polynomial of a simple polynomial. 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 simplicial complex arising from the triangulation can be computed via a shelling of the complex. 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 major index and inversion index are shown to be equidistributed over S n d. Likewise, the pair of statistics (d; maj) is shown to be equidistributed with the pair (; den), where den is Denert's statistic and is an alternative deenition of excedance. A result of Stanley, relating the number of permutations with k descents to the volume of a certain \slice" of the unit d{cube, is also generalized.