Related papers: The Weak Bruhat Order and Separable Permutations
Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author…
A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic…
This article studies the poset of simple permutations with respect to the pattern involvement. We specify results on critically indecomposable posets obtained by Schmerl and Trotter to simple permutations and prove that if $\sigma, \pi$ are…
We investigate the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. We first prove a strong version of a conjecture by Castiglione and…
We construct the algebra of fractions of a Weak Bialgebra relative to a suitable denominator set of group-like elements that is `almost central', a condition we introduce in the present article which is sufficient in order to guarantee…
Given a permutation $\pi\in \Sn\_n$, construct a graph $G\_\pi$ on the vertex set $\{1,2, ..., n\}$ by joining $i$ to $j$ if (i) $i<j$ and $\pi(i)<\pi(j)$ and (ii) there is no $k$ such that $i < k < j$ and $\pi(i)<\pi(k)<\pi(j)$. We say…
Let $S_n$ denote the group all permutations of $n$. For every permutation $\sigma$, we let $\mathrm{des}(\sigma)$ denote the number of descents in $\sigma$ and $\mathrm{LRMin}(\sigma)$ denote the number of left-to-right minima of $\sigma$.…
Given a Coxeter system (W,S) equipped with an involutive automorphism T, the set of twisted identities is i(T) = {T(w)^{-1}w : w \in W}. We point out how i(T) shows up in several contexts and prove that if there is no s \in S such that…
Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function…
The gap probability generating function has as its coefficients the probability of an interval containing exactly $k$ eigenvalues. For scaled random matrices with orthogonal symmetry, and the interval at the hard or soft spectrum edge, the…
The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M\"obius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the M\"obius function…
We continue the study of separable elements in finite Weyl groups. These elements generalize the well-studied class of separable permutations. We show that the multiplication map $W/U \times U \to W$ is a length-additive bijection, or…
This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to…
Recently Albert and Bousquet-M\'elou \cite{AB15} obtained the solution to the long-standing problem of the number of permutations sortable by two stacks in parallel (tsip). Their solution was expressed in terms of functional equations. We…
The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these…
We present several multi-variable generating functions for a new pattern matching condition on the wreath product of the cyclic group and the symmetric group. Our new pattern matching condition requires that the underlying permutations…
This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called…
In this paper, we define a set which has a finite group action and is generated by a finite color set, a set which has a finite group action, and a subset of the set of non negative integers. we state its properties to apply one of solution…
In this paper, we construct the weak version of peak quasisymmetric functions inside the Hopf algebra of weak composition quasisymmetric functions (WCQSym) defined by Guo, Thibon and Yu. Weak peak quasisymmetric functions (WPQSym) are…
This is the first contribution of a sequence of papers introducing the notions of $s$-weak order and $s$-permutahedra, certain discrete objects that are indexed by a sequence of non-negative integers $s$. In this first paper, we concentrate…