In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the -diagrams of Alex Postnikov [A. Postnikov, Webs in totally positive Grassmann cells, in preparation; L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative sizes of pairs of letters in a permutation and their places. We call these statistics weak excedance statistics because of their close relation to weak excedances. The second bijection carries tableaux statistics (via the weak excedance statistics) to statistics based on generalized permutation patterns. We then give enumerative applications of these bijections. One nice consequence of these results is that the polynomial enumerating permutation tableaux according to their content generalizes both Carlitz' q-analog of the Eulerian numbers [L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332-350] and the more recent q-analog of the Eulerian numbers found in [L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. We conclude our paper with a list of open problems, as well as remarks on progress on these problems which has been made by A. Burstein, S. Corteel, N. Eriksen, A. Reifegerste, and X. Viennot.

We define new Mahonian statistics, called MAD, MAK, and ENV, on words. Of these, ENV is shown to equal the classical INV, that is, the number of inversions, while for permutations MAK has been already defined by Foata and Zeilberger. It Ž . Ž . is shown that the triple statistics des, MAK, MAD and exc, DEN, ENV are equidistributed over the rearrangement class of an arbitrary word. Here, exc is the number of excedances and DEN is Denert's statistic. In particular, this implies the Ž . Ž . equidistribution of exc, INV and des, MAD . These bistatistics are not equidis-Ž . tributed with the classical Euler᎐Mahonian statistic des, MAJ . The proof of the main result is by means of a bijection which, in the case of permutations, is Ž . essentially equivalent to several bijections in the literature or inverses of these .

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.

We introduce a new combinatorial object called a web world that consists of a set of web diagrams. The diagrams of a web world are generalizations of graphs, and each is built on the same underlying graph. Instead of ordinary vertices the diagrams have pegs, and edges incident to a peg have different heights on the peg. The web world of a web diagram is the set of all web diagrams that result from permuting the order in which endpoints of edges appear on a peg. The motivation comes from particle physics, where web diagrams arise as particular types of Feynman diagrams describing scattering amplitudes in non-Abelian gauge (Yang-Mills) theories. To each web world we associate two matrices called the web-colouring matrix and web-mixing matrix. The entries of these matrices are indexed by ordered pairs of web diagrams (D1, D2), and are computed from those colourings of the edges of D1 that yield D2 under a transformation determined by each colouring.We show that colourings of a web diagram (whose constituent indecomposable diagrams are all unique) that lead to a reconstruction of the diagram are equivalent to order-preserving mappings of certain partially ordered sets (posets) that may be constructed from the web diagrams. For web worlds whose web graphs have all edge labels equal to 1, the diagonal entries of web-mixing and web-colouring matrices are obtained by summing certain polynomials determined by the descents in permutations in the Jordan-Hölder set of all linear extensions of the associated poset. We derive tri-variate generating generating functions for the number of web worlds according to three statistics and enumerate the number of different web diagrams in a web world. Three special web worlds are examined in great detail, and the traces of the web-mixing matrices calculated in each case.