Permutation Statistics of Indexed Permutations

Abstract

The definitions of descent, excedance, major index, inversion index and Denert's statistic for the elements of the symmetric group S d are generalized to indexed permutations, i.e. the elements of the group S n d := Z n ≀ S d , where ≀ is wreath product with respect to the usual action of S d by permutations 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 hpolynomial (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≥0 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 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 S n d. Likewise, the pair of statistics (d, maj) is shown to be equidistributed with the pair (ǫ, den), where den is Denert's statistic and ǫ is an alternative definition 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.

Proof of a Conjecture of Ehrenborg and Steingrı́msson on Excedance Statistic

“…Relevant results to permutation statistics and log-concavity of sequences see [2][3][4][5][6][8][9][10][11][12][13][14].…”

Mixed Volumes and Slices of the Cube

“…The set of all such indexed permutations is denoted by S For the rest of this note, let * be a non-negative integer and n=*+1. The following is proved in [11,Theorem 50]. …”

“…K An indexed permutation of length d and with indices in [0, 1, ..., n&1] is an ordinary permutation in the symmetric group S d where each letter has been assigned an integer between 0 and n&1. Indexed permutations, or r-signed permutations, are a generalization of permutations; see [1,2,11]. We will follow the notation in [11].…”

“…Indexed permutations, or r-signed permutations, are a generalization of permutations; see [1,2,11]. We will follow the notation in [11]. The set of all such indexed permutations is denoted by S For the rest of this note, let * be a non-negative integer and n=*+1.…”

“…Proposition 3 [11]. The volume of the slice X k is equal to the number of indexed permutations in S n d with k descents.…”