Related papers: Counting factorizations of Singer cycles in linear…
The number of shortest factorizations into reflections for a Singer cycle in GL_n(F_q) is shown to be (q^n-1)^(n - 1). Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also…
In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is…
We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings; that is, we study the number of ways a permutation can be decomposed into a product of a given number of 2-cycles, 3-cycles, etc.…
It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of $k$ cycles of given lengths has a very simple formula: it is $n^{k-1}$ where $n$ is the rank of the underlying symmetric group…
We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our…
We give a short proof, based on symmetric function theory, of a formula due to Goupil and Schaeffer, counting the number of factorizations of a cycle of maximal length in the symmetric group, into the product of two permutations of given…
We consider the determination of the number $c_k(\alpha)$ of ordered factorisations of an arbitrary permutation on n symbols, with cycle distribution $\alpha$, into k-cycles such that the factorisations have minimal length and such that the…
In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection…
We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of…
We give an intrinsic criterion to tell whether a reflection factorization in the general linear group is reduced, and give a formula for computing reflection length in the general affine group.
We develop first steps in the study of factorizations of elements in ultraproducts of commutative cancellative monoids into irreducible elements. A complete characterization of the (multi-)sets of lengths in such objects is given. As…
Two factorizations of a permutation into products of cycles are equivalent if one can be obtained from the other by repeatedly interchanging adjacent disjoint factors. This paper studies the enumeration of equivalence classes under this…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
This paper is devoted to the computation of the number of ordered factorizations of a long cycle in the symmetric group where the number of factors is arbitrary and the cycle structure of the factors is given. Jackson (1988) derived the…
We give a new expression for the number of factorizations of a full cycle into an ordered product of permutations of specified cycle types. This is done through purely algebraic means, extending work of Biane. We deduce from our result a…
In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which…
Factorization of string amplitudes is one way of constructing string interaction vertices. We show that correlation functions in string theory can be conveniently factorized using loop variables representing delta functionals. We illustrate…
The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for…
In this paper, we enumerate the pairs of permutations that are long cycles and whose product has a given cycle-type. Our main result is a simple relation concerning the desired numbers for a few related cycle-types. The relation refines a…
We consider GL_n(F_q)-analogues of certain factorization problems in the symmetric group S_n: rather than counting factorizations of the long cycle (1, 2, ..., n) given the number of cycles of each factor, we count factorizations of a…