Permutation Statistics of Indexed Permutations

Abstract

The definitions of descent, excedance, major index, inversion index and Denert's statistics for the elements of the symmetric group Ld are generalized to indexed permutation, i.e. the elements of the group Snd: = Zn ≀ Ld, where ≀ is the wreath product with respect to the usual action of Ld by permutation of {1, 2,…, d}.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 ∑d ⩾ 0Edxd /d! = sec x + tan x, where Ed is the number of alternating permutations in Ld, is generalized in two different 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 Snd . Likewise, the pair of statistics (d, maj) is shown to be equidistributed with the pair (ε, den), where den is Denert's statistic and ε is an alternative definition of excedance. A result relating the number of permutations with k descents to the volume of a certain 'slice' of the unit d-cube is also generalized.