On comparability of bigrassmannian permutations

Let Sn and Gn denote the respective sets of ordinary and bigrassmannian (BG) permutations of order n, and let (Gn,≤) denote the Bruhat ordering permutation poset. We study the restricted poset (Bn,≤), first providing a simple criterion for comparability. This criterion is used to show that that the poset is connected, to enumerate the saturated chains between elements, and to enumerate the number of maximal elements below r fixed elements. It also quickly produces formulas for β(ω) (α(ω), respectively), the number of BG permutations weakly below (weakly above, respectively) a fixed ω ∈ Bn, and is used to compute the Mo¨bius function on any interval in Bn. We then turn to a probabilistic study of β = β(ω) (α = α(ω) respectively) for the uniformly random ω ∈ Bn. We show that α and β are equidistributed, and that β is of the same order as its expectation with high probability, but fails to concentrate about its mean. This latter fact derives from the limiting distribution of β/n3. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain

The coincidence of the Bruhat order and the secondary Bruhat order on A(n,k)\mathcal{A}(n,k)

Given a positive integer $n$ and a nonnegative integer $k$ with $k\leq n$, we denote by $\mathcal{A}(n,k)$ the class of all $n$-by-$n$ $(0,1)$-matrices with constant row and column sums $k$. In this paper, we show that the Bruhat order and the secondary Bruhat order coincide on $\mathcal{A}(n,k)$ if and only if either $0\leq n\leq 5$ or $k\in\{0,1,2,n-2,n-1,n\}$ with $n\geq 6$.Comment: 8 page

On the little secondary bruhat order

CMA and Departamento de Matematica da Faculdade de Ciencias Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal.([email protected]).Partially supported by the Fundacao para a Ciencia e a Tecnologia through the project UID/MAT/04721/2019. Departamento de Matematica da Universidade da Beira Interior, Rua Marques D'Avila and Bolama, 6201-001 Covilha, Portugal ([email protected]).Partially supported by the Fundacao para a Ciencia e a Tecnologia through the project UIDB/MAT/00212/2020.Let R and S be two sequences of positive integers in nonincreasing order having the same sum. We denote by A(R, S) the class of all (0, 1)-matrices having row sum vector R and column sum vector S. Brualdi and Deaett (More on the Bruhat order for (0, 1)-matrices, Linear Algebra Appl., 421:219{232, 2007) suggested the study of the secondary Bruhat order on A(R, S) but with some constraints. In this paper, we study the cover relation and the minimal elements for this partial order relation, which we call the little secondary Bruhat order, on certain classes A(R, S). Moreover, we show that this order is different from the Bruhat order and the secondary Bruhat order. We also study a variant of this order on certain classes of symmetric matrices of A(R, S).publishersversionpublishe

On the Sperner property for the absolute order on complex reflection groups

Two partial orders on a reflection group, the codimension order and the prefix order, are together called the absolute order when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner property, except possibly for the Coxeter group of type $D_n$, for which this property is conjectural. The Sperner property had previously been established for the noncrossing partition lattice $NC_W$, a certain maximal interval in the absolute order, but not for the entire poset, except in the case of the symmetric group. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.Comment: 12 pages, comments welcome; v2: minor edits and journal referenc

Toward the Enumeration of Maximal Chains in the Tamari Lattices

abstract: The Tamari lattices have been intensely studied since they first appeared in Dov Tamari’s thesis around 1952. He defined the n-th Tamari lattice T(n) on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Despite their interesting aspects and the attention they have received, a formula for the number of maximal chains in the Tamari lattices is still unknown. The purpose of this thesis is to convey my results on progress toward the solution of this problem and to discuss future work. A few years ago, Bergeron and Préville-Ratelle generalized the Tamari lattices to the m-Tamari lattices. The original Tamari lattices T(n) are the case m=1. I establish a bijection between maximum length chains in the m-Tamari lattices and standard m-shifted Young tableaux. Using Thrall’s formula, I thus derive the formula for the number of maximum length chains in T(n). For each i greater or equal to -1 and for all n greater or equal to 1, I define C(i,n) to be the set of maximal chains of length n+i in T(n). I establish several properties of maximal chains (treated as tableaux) and identify a particularly special property: each maximal chain may or may not possess a plus-full-set. I show, surprisingly, that for all n greater or equal to 2i+4, each member of C(i,n) contains a plus-full-set. Utilizing this fact and a collection of maps, I obtain a recursion for the number of elements in C(i,n) and an explicit formula based on predetermined initial values. The formula is a polynomial in n of degree 3i+3. For example, the number of maximal chains of length n in T(n) is n choose 3. I discuss current work and future plans involving certain equivalence classes of maximal chains in the Tamari lattices. If a maximal chain may be obtained from another by swapping a pair of consecutive edges with another pair in the Hasse diagram, the two maximal chains are said to differ by a square move. Two maximal chains are said to be in the same equivalence class if one may be obtained from the other by making a set of square moves.Dissertation/ThesisDoctoral Dissertation Mathematics 201

Antichains in weight posets associated with gradings of simple Lie algebras

For a reductive Lie algebra $\mathfrak h$ and a simple finite-dimensional $\mathfrak h$-module $V$, the set of weights of $V$, $P(V)$, has a natural poset structure. We consider antichains in the weight poset $P(V)$ and a certain operator $\mathfrak X$ acting on antichains. Eventually, we impose stronger constraints on $(\mathfrak h,V)$ and stick to the case in which $\mathfrak h$ and $V$ are associated with a $Z$-grading of a simple Lie algebra $\mathfrak g$. Then $V$ is a weight multiplicity free $\mathfrak h$-module and $P(V)$ can be regarded as a subposet of $\Delta^+$, where $\Delta$ is the root system of $\mathfrak g$. Our goal is to demonstrate that antichains in the weight posets associated with $Z$-gradings of $\mathfrak g$ exhibit many good properties similar to those of $\Delta^+$ that are observed earlier in arXiv: math.CO 0711.3353 (=Ref. [14] in the text).Comment: 28 page