Signed mahonians on some trees and parabolic quotients

We study the distribution of the major index with sign on some parabolic quotients of the symmetric group, extending and generalizing simultaneously results Gessel-Simion and Adin-Gessel-Roichman, and on some special trees that we call rakes. We further consider and compute the distribution of the flag-major index on some parabolic quotients of wreath products and other related groups. All these distributions turn out to have very simple factorization formulas.Comment: 12 page

Generalizations of Permutation Statistics to Words and Labeled Forests

A classical result of MacMahon shows the equidistribution of the major index and inversion number over the symmetric groups. Since then, these statistics have been generalized in many ways, and many new permutation statistics have been defined, which are related to the major index and inversion number in may interesting ways. In this dissertation we study generalizations of some newer statistics over words and labeled forests. Foata and Zeilberger defined the graphical major index, majU , and the graphical inversion index, invU , for words over the alphabet {1, . . . , n}. In this dissertation we define a graphical sorting index, sorU , which generalizes the sorting index of a permutation. We then characterize the graphs U for which sorU is equidistributed with invU and majU on a single rearrangement class. Bj¨orner and Wachs defined a major index for labeled plane forests, and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests. Specifically, we introduce the notions of bottom-to-top maxima, cyclic bottom-to-top maxima, sorting index, and cycle minima. Then we show that the pairs (inv, BT-max), (sor, Cyc), and (maj, CBT-max) are equidistributed. Our results extend the result of Bj¨orner and Wachs and generalize results for permutations. Lastly, we study the descent polynomial of labeled forests. The descent polynomial for per-mutations is known to be log-concave and unimodal. In this dissertation we discuss what properties are preserved in the descent polynomial of labeled forests

q-Hook length formulas for colored labeled forests

The major index has been deeply studied from the early 1900s and recently has been generalized in different directions, such as the case of labeled forests and colored permutations. In this thesis we define new types of labelings for forests in which the labels are colored integers. We extend the definition of the flag-major index for these labelings and we present an analogue of well known major index hook length formulas. Finally, this study (which has just apparently a simple combinatoric nature) allows us to show a notion of duality for two particular families of groups obtained from the product G(r,n)×G(r,m)

Coxeter factorizations with generalized Jucys-Murphy weights and Matrix Tree theorems for reflection groups

We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group $W$, in terms of the spectrum of an associated operator, the $W$-Laplacian. This covers in particular all finite Coxeter groups. The results of this paper include generalizations of the Matrix Tree and Matrix Forest theorems to reflection groups, and cover reduced (shortest length) as well as arbitrary length factorizations. Our formulas are relative to a choice of weighting system that consists of $n$ free scalar parameters and is defined in terms of a tower of parabolic subgroups. To study such systems we introduce (a class of) variants of the Jucys-Murphy elements for every group, from which we define a new notion of `tower equivalence' of virtual characters. A main technical point is to prove the tower equivalence between virtual characters naturally appearing in the problem, and exterior products of the reflection representation of $W$. Finally we study how this $W$-Laplacian matrix we introduce can be used in other problems in Coxeter combinatorics. We explain how it defines analogues of trees for $W$ and how it relates them to Coxeter factorizations, we give new numerological identities between the Coxeter number of $W$ and those of its parabolic subgroups, and finally, when $W$ is a Weyl group, we produce a new, explicit formula for the volume of the corresponding root zonotope.Comment: 57 pages, comments are very much welcom