相关论文: Signed mahonians on some trees and parabolic quoti…
We study the generating polynomial of the flag major index with each one-dimensional character, called signed Mahonian polynomial, over the colored permutation group, the wreath product of a cyclic group with the symmetric group. Using the…
A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was…
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…
We relate properties of weighted flags (or multiflags) of type AD to statistics of the corresponding Weyl groups. For type A, we recover the Mahonian statistics on symmetric groups. Finally, we sketch briefly an easy extension incorporating…
A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type $B$, it is equidistributed with length. For more general wreath products it appears in an explicit formula for…
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.…
We derive some new signed Mahonian polynomials over the complex reflection group $G(r,1,n)=C_r\wr\mathfrak{S}_n$, where the "sign" is taken to be any of the $2r$ $1$-dim characters and the "Mahonian" statistics are the $\mathsf{lmaj}$…
In 1997 Clarke et al. studied a $q$-analogue of Euler's difference table for $n!$ using a key bijection $\Psi$ on symmetric groups. In this paper we extend their results to the wreath product of a cyclic group with the symmetric group. In…
We consider conjugation action of symmetric group on the semigroup of all partial functions and develop a machinery to investigate character formulas and multiplicities. In particular, we determine nilpotent matrices whose orbit under…
We develop direct bijections between the set $F_n^k$ of minimal factorizations of the long cycle $(0\,1\,\cdots\, kn)$ into $(k+1)$-cycle factors and the set $R_n^k$ of rooted labelled forests on vertices $\{1,\ldots,n\}$ with edges…
We introduce the Major MacMahon map and show how this map interacts with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the ab-index of a simplicial poset, it yields the q-analogue of n! times the…
We consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We use cumulants to classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary…
The Sackin index is an important measure for the balance of phylogenetic trees. We investigate two extensions of the Sackin index to the class of galled trees and two of its subclasses (simplex galled trees and normal galled trees) where we…
We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes…
We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that…
Recently Cheng et al. (Adv. in Appl. Math. 143 (2023) 102451) generalized the inversion number to partial permutations, which are also known as Laguerre digraphs, and asked for a suitable analogue of MacMahon's major index. We provide such…
MacMahon's classic theorem states that the 'length' and 'major index' statistics are equidistributed on the symmetric group S_n. By defining natural analogues or generalizations of those statistics, similar equidistribution results have…
Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr \mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. We prove that the irreducible representations for this class of groups are indexed by a…
We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian…
In this paper, we construct a mixed-base number system over the generalized symmetric group $G(m,1,n)$, which is a complex reflection group with a root system of type $B_n^{(m)}$. We also establish one-to-one correspondence between all…