We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation σ = σ1σ2 · · · σn defined as the set of indices i such that either i is odd and σi > σi+1, or i is even and σi < σi+1. We show that this statistic is equidistributed with the 3-descent set statistic on permutationsσ = σ1σ2 · · · σn+1 with σ1 = 1, defined to be the set of indices i such that the triple σiσi+1σi+2 forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials P σ∈Sn t des(σ)+1 using alternating descents. For the alternating descent set statistic, we define the generating polynomial in two non-commutative variables by analogy with the ab-index of the Boolean algebra Bn, and make observations about it. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new q-analog of the Euler number En and show how it emerges in a q-analog of an identity expressing En as a weighted sum of Dyck paths.

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up notatio

We consider the inversion enumerator In(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = −1 into this generalized polynomial produces the number of permutations with a certain descent set. In the classical case, this result implies the formula In(−1) = En, the number of alternating permutations. We give a combinatorial proof of these formulas based on the involution principle. We also give a geometric interpretation of these identities in terms of volumes of generalized chain polytopes of ribbon posets. The volume of such a polytope is given by a sum over generalized parking functions, which is similar to an expression for the volume of the parking function polytope of Pitman and Stanley. q b 1 +b 2 +···+bn−n = q ( n 2 ) · I n (q −1 ), where P n is the set of all parking functions of length n. Cayley's formula states that |T n | = |P n | = (n + 1) n−1 , hence I n (1) = (n + 1) n−1 .

For a positive integer n and a subset S ⊆ [n − 1], the descent polytope DP S is the set of points (x 1 , . . . , x n ) in the n-dimensional unit cube [0, 1] n such that x i ≥ x i+1 if i ∈ S and x i ≤ x i+1 otherwise. First, we express the f -vector as a sum over all subsets of [n − 1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f -vector. We show that the f -vector is maximized when the set S is the alternating set {1, 3, 5, . . .} ∩ [n − 1]. We derive a generating function for F S (t), written as a formal power series in two non-commuting variables with coefficients in Z[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.

A graph is k-ordered if, for any sequence of k vertices, there is a cycle containing these vertices in the given order. A graph is k-edge-ordered if, for any sequence of k edges, there is a tour containing these edges in the given order. Finally, a graph is strongly k-edge-ordered if for any sequence of k oriented edges, there is a tour containing these edges in the given order and in the given orientations. In this paper, we prove that every 2k-ordered (resp. (2k + 1)-ordered) graph is k-edge-ordered (resp. strongly k-edge-ordered). We also examine degree conditions and connectivity for k-edge-ordered graphs, and state results on k-edge-ordered Eulerian graphs.