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The distribution of Coxeter descents and block number over the set of fully commutative elements in the hyperoctahedral group $B_n$, $\FC(B_n)$, is studied in this paper. We prove that the associated Chow quasi-symmetric generating function is equal to a non-negative sum of products of two Schur functions. The proof involves a decomposition of $\FC(B_n)$ into a disjoint union of two-sided Barbash-Vogan combinatorial cells, a type $B$ extension of Rubey's descent preserving involution on $321$-avoiding permutations and a detailed study of the intersection of $\FC(B_n)$ with $S_n$-cosets which yields a new decomposition of $\FC(B_n)$ into disjoint subsets called fibers. We also compare two different type $B$ Schur-positivity notions, arising from works of Chow and Poirier, 25 pages, 6 figures

Ce mémoire constitue un travail de synthèse de mes travaux dans le domaine de la combinatoire énumérative et algébrique autour des groupes de permutations généralisées : ce terme désignera ici les groupes de Coxeter classiques et certains groupes de réflexions complexes. La plupart des résultats présentés dans ce mémoire peuvent être classifiés en deux catégories. Il s'agit soit de résultats algébriques dont nous donnons des descriptions combinatoires explicites, soit de résultats énumératifs qui ont une signification dans un contexte algébrique particulier. Pour faire cela, on s'appuie souvent sur des fonctions particulières à valeurs entières positives qui sont usuellement appelées statistiques de permutations.

International audience; We give a new description of the flag major index, introduced by Adin and Roichman, by using a major index defined by Reiner. This allows us to establish a connection between an identity of Reiner and some more recent results due to Chow and Gessel. Furthermore we generalize the main identity of Chow and Gessel by computing the four-variate generating series of descents, major index, length, and number of negative entries over Coxeter groups of type $B$ and $D$.

We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics $(\des_G, \maj,\ell_G, \col)$ and $(\des_G, \ides_G, \maj, \imaj, \col, \icol)$ over the wreath product of a symmetric group by a cyclic group. Here $\des_G$, $\ell_G$, $\maj$, $\col$, $\ides_G$, $\imaj_G$, 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 $B$., 19 pages

We find a minimal generating set for the De Concini-Procesi ideals indexed by hooks, and study their minimal free resolutions as well as their Hilbert series and regularity., 20 pages

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., 28 pages, 12 figures

Using Reiner's definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $R[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$., 17 pages, 2 figures; submitted