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We study $321$-avoiding affine permutations, and prove a formula for their enumeration with respect to the inversion number by using a combinatorial approach. This is done in two different ways, both related to Viennot's theory of heaps. First, we encode these permutations using certain heaps of monomers and dimers. This method specializes to the case of affine involutions. For the second proof, we introduce periodic parallelogram polyominoes, which are new combinatorial objects of independent interest. We enumerate them by extending the approach of Bousquet-M\'elou and Viennot used for classical parallelogram polyominoes. We finally establish a connection between these new objects and $321$-avoiding affine permutations., Comment: 25 pages, 17 figures

We give exact formulas for the bivariate generating series of 321-avoiding affine permutations with respect to rank and Coxeter length. We use two different combinatorial approaches, both based on the theory of heaps of pieces.

An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennot's heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine type $A$, we connect our results to Fan--Green's cell structure of the corresponding Temperley--Lieb algebra., 25 pages

We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics (des(G), maj, l(G), col) and (des(G), ides(G), maj, imaj, col, icol) over the wreath product of a symmetric group by a cyclic group. Here desG, l(G), maj, col, idesG, imaj(G), and icol denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, and Reiner on various distributions of statistics over Coxeter groups of type A and B.

We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes. Natural Sciences and Engineering Research Council of Canada Fonds Québécois de la Recherche sur la Nature et les Technologies

The generating functions of the major index and of the flag-major index, with each of the one-dimensional characters over the symmetric and hyperoctahedral group, respectively, have simple product formulas. In this paper, we give a factorial-type formula for the generating function of the D-major index with sign over the Weyl groups of type D. This completes a picture which is now known for all the classical Weyl groups., Comment: 14 pages

Given a classical Weyl group $W$, that is, a Weyl group of type $A$, $B$ or $D$, one can associate with it a polynomial with integral coefficients $Z_W$ given by the ratio of the Hilbert series of the invariant algebras of the natural action of $W$ and $W^t$ on the ring of polynomials ${\bf C}[x_1, \ldots , x_n]^{\otimes t}$. We introduce and study several statistics on the classical Weyl groups of type $B$ and $D$ and show that they can be used to give an explicit formula for $Z_{D_n}$. More precisely, we define two Mahonian statistics, that is, statistics having the same distribution as the length function, $Dmaj$ and $ned$ on $D_n$. The statistic $Dmaj$, defined in a combinatorial way, has an analogous algebraic meaning to the major index for the symmetric group and the flag-major index of Adin and Roichman for $B_n$; namely, it allows us to find an explicit formula for $Z_{D_n}$. Our proof is based on the theory of $t$-partite partitions introduced by Gordon and further studied by Garsia and Gessel. Using similar ideas, we define the Mahonian statistic $ned$ also on $B_n$ and we find a new and simpler proof of the Adin?Roichman formula for $Z_{B_n}$. Finally, we define a new descent number $Ddes$ on $D_n$ so that the pair $(Ddes,Dmaj)$ gives a generalization to $D_n$ of the Carlitz identity on the Eulerian?Mahonian distribution of descent number and major index on the symmetric group.

We introduce and study three new statistics on the even-signed permutation group D_n. We show that two of these are Mahonian, i.e., are equidistributed with length, and that a pair of them gives a generalization of Carlitz’s identity on the Euler-Mahonian distribution of the descent number and major index over S_n.

A monomial basis for the coinvariant algebra of type D is introduced. This basis naturally leads to the definition of a new family of representations that decompose the homogeneous components of the coinvariant algebra. Furthermore, the multiplicity of the irreducible characters within the characters of these representations is described in terms of tableaux. This algebraic setting is then applied to generalize various combinatorial identities and find new ones.