scholarly journals A Combinatorial Formula for Kazhdan-Lusztig Polynomials of $\rho$-Removed Uniform Matroids

10.37236/9435 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Kyungyong Lee ◽  
George D. Nasr ◽  
Jamie Radcliffe
Keyword(s):  
Positive Integer ◽  
Negative Integer ◽  
Young Tableaux ◽  
Combinatorial Formula ◽  
Integer Coefficients ◽  
Lusztig Polynomials

Let $\rho$ be a non-negative integer. A $\rho$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $\rho$ disjoint bases. We present a combinatorial formula for Kazhdan–Lusztig polynomials of $\rho$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.

scholarly journals The Coin Exchange Problem and the Structure of Cube Tilings

10.37236/2251 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Andrzej Piotr Kisielewicz ◽  
Krzysztof Przeslawski
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Unit Cube ◽  
Negative Integer ◽  
Integer Coefficients ◽  
Cube Tiling ◽  
Cube Tilings

It is shown that if $[0,1)^d+t$, $t\in T$, is a unit cube tiling of $\mathbb{R}^d$, then for every $x\in T$, $y\in \mathbb{R}^d$, and every positive integer $m$ the number $|T\cap (x+\mathbb{Z}^d)\cap([0,m)^d+ y)|$ is divisible by $m$. Furthermore, by a result of Coppersmith and Steinberger on cyclotomic arrays it is proven that for every finite discrete box $D=D_1\times\cdots\times D_d \subseteq x+\mathbb{Z}^d$ of size $m_1\times \cdots\times m_d$ the number $|D\cap T|$ is a linear combination of $m_1,\ldots, m_d$ with non-negative integer coefficients. Several consequences  are collected. A generalization is presented.      

scholarly journals A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Kyungyong Lee ◽  
George D. Nasr ◽  
Jamie Radcliffe
Keyword(s):  
Young Tableaux ◽  
Combinatorial Formula ◽  
Combinatorial Interpretation ◽  
Special Cases ◽  
Almost All ◽  
Lusztig Polynomials

We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan-Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.

scholarly journals Deodhar Elements in Kazhdan-Lusztig Theory

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Brant Jones
Keyword(s):  
Coxeter Groups ◽  
Pattern Avoidance ◽  
Weyl Groups ◽  
Schubert Varieties ◽  
Combinatorial Formula ◽  
Nous Obtenons ◽  
Integer Coefficients ◽  
Nous Montrons ◽  
International Audience ◽  
Lusztig Polynomials

International audience The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered. Les polynômes de Kazhdan-Lusztig $P_{x,w}(q)$ des groupes de Weyl finis apparaissent en théorie des représentations, ainsi qu’en géométrie des variétés de Schubert. Il a été démontré peu après leur introduction qu’ils avaient des coefficients entiers positifs, mais on ne connaît toujours pas d’interprétation combinatoire simple de cette propriété dans le cas général. Deodhar a proposé un cadre donnant un algorithme, en général récursif, calculant des formules attractives pour les polynômes de Kazhdan-Lusztig. Billey-Warrington ont démontré que cet algorithme est non récursif lorsque$w$ évite les hexagones et les $321$ et qu’il donne des formules combinatoires simples. Nous introduisons une notion d’évitement de schémas dansles groupes de Coxeter quelconques nous permettant de généraliser les résultats de Billey-Warrington à tout groupe de Weyl fini. Nous montrons que le coefficient de tête $\mu (x,w)$ de ces polynômes de Kazhdan-Lusztig est toujours $0$ ou $1$. Cela généralise aussi des résultats de Fan-Greenqui identifient les groupes de Coxeter complètement serrés. Enfin, en type $A$, nous obtenons une classe plus large de permutations évitant la récursion.

scholarly journals On Plücker coordinates of a perfectly oriented planar network

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Kelli Talaska
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Weight Functions ◽  
Combinatorial Formula ◽  
Rational Map ◽  
Positive Weight ◽  
Integer Coefficients ◽  
Plücker Coordinates ◽  
Planar Network ◽  
International Audience

International audience Let $G$ be a perfectly oriented planar graph. Postnikov's boundary measurement construction provides a rational map from the set of positive weight functions on the edges of $G$ onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov's map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weights, with positive integer coefficients. These polynomials are weight generating functions for certain subsets of edges in $G$.

scholarly journals Reachability Relations and the Structure of Transitive Digraphs

10.37236/115 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Norbert Seifter ◽  
Vladimir I. Trofimov
Keyword(s):  
Positive Integer ◽  
Negative Integer ◽  
Equivalence Relations ◽  
Basic Relation ◽  
Factor Graphs ◽  
Locally Finite ◽  
General Properties

In this paper we investigate reachability relations on the vertices of digraphs. If $W$ is a walk in a digraph $D$, then the height of $W$ is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed opposite to their orientation. Two vertices $u,v\in V(D)$ are $R_{a,b}$-related if there exists a walk of height $0$ between $u$ and $v$ such that the height of every subwalk of $W$, starting at $u$, is contained in the interval $[a,b]$, where $a$ ia a non-positive integer or $a=-\infty$ and $b$ is a non-negative integer or $b=\infty$. Of course the relations $R_{a,b}$ are equivalence relations on $V(D)$. Factorising digraphs by $R_{a,\infty}$ and $R_{-\infty,b}$, respectively, we can only obtain a few different digraphs. Depending upon these factor graphs with respect to $R_{-\infty,b}$ and $R_{a,\infty}$ it is possible to define five different "basic relation-properties" for $R_{-\infty,b}$ and $R_{a,\infty}$, respectively. Besides proving general properties of the relations $R_{a,b}$, we investigate the question which of the "basic relation-properties" with respect to $R_{-\infty,b}$ and $R_{a,\infty}$ can occur simultaneously in locally finite connected transitive digraphs. Furthermore we investigate these properties for some particular subclasses of locally finite connected transitive digraphs such as Cayley digraphs, digraphs with one, with two or with infinitely many ends, digraphs containing or not containing certain directed subtrees, and highly arc transitive digraphs.

scholarly journals On the Number of Partition Weights with Kostka Multiplicity One

10.37236/2574 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Zachary Gates ◽  
Brian Goldman ◽  
C. Ryan Vinroot
Keyword(s):  
Positive Integer ◽  
Ordered Set ◽  
Unitary Groups ◽  
Character Theory ◽  
Partition Functions ◽  
Young Tableaux ◽  
Multiplicity One ◽  
Restricted Partition ◽  
Finite Unitary Groups ◽  
Kostka Number

Given a positive integer $n$, and partitions $\lambda$ and $\mu$ of $n$, let $K_{\lambda \mu}$ denote the Kostka number, which is the number of semistandard Young tableaux of shape $\lambda$ and weight $\mu$.  Let $J(\lambda)$ denote the number of $\mu$ such that $K_{\lambda \mu} = 1$.  By applying a result of Berenshtein and Zelevinskii, we obtain a formula for $J(\lambda)$ in terms of restricted partition functions, which is recursive in the number of distinct part sizes of $\lambda$.  We use this to classify all partitions $\lambda$ such that $J(\lambda) = 1$ and all $\lambda$ such that $J(\lambda) = 2$.  We then consider signed tableaux, where a semistandard signed tableau of shape $\lambda$ has entries from the ordered set $\{0 < \bar{1} < 1 < \bar{2} < 2 < \cdots \}$, and such that $i$ and $\bar{i}$ contribute equally to the weight.  For a weight $(w_0, \mu)$ with $\mu$ a partition, the signed Kostka number $K^{\pm}_{\lambda,(w_0, \mu)}$ is defined as the number of semistandard signed tableaux of shape $\lambda$ and weight $(w_0, \mu)$, and $J^{\pm}(\lambda)$ is then defined to be the number of weights $(w_0, \mu)$ such that $K^{\pm}_{\lambda, (w_0, \mu)} = 1$.  Using different methods than in the unsigned case, we find that the only nonzero value which $J^{\pm}(\lambda)$ can take is $1$, and we find all sequences of partitions with this property.  We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

Multidimensional right-shift processes

1971 ◽  
Vol 3 (02) ◽  
pp. 200-201 ◽  
Author(s):  
Norman C. Severo
Keyword(s):  
Positive Integer ◽  
Random Variable ◽  
Negative Integer

Let v be a positive integer and for each k = 1, · · ·, v let m k and N k be a positive and a non-negative integer, respectively. Denote by S'N k ,m k the set of (m k + 1) -tuples r k = (r k ,m k , · · ·, r k,1, r k,0) having non-negative components summing to N k , and by X k (t) = (X k,m k (t), · · ·, X k,1(t), X k ,0(t)) an (m k + 1)-tuple random variable taking on values only from the set S′ N k ,m k .

A physical model for teaching multiplication of integers

1968 ◽  
Vol 15 (6) ◽  
pp. 525-528
Author(s):  
Warren H. Hill
Keyword(s):  
Positive Integer ◽  
Physical Model ◽  
Number Line ◽  
Negative Integer ◽  
Physical Models ◽  
Addition And Subtraction ◽  
Positive Integers ◽  
Partial Success

One of the pedagogical pitfalls involved in teaching the multiplication of integers is the scarcity of physical models that can be used to illustrate their product. This problem becomes even more acute if a phyiscal model that makes use of a number line is desired. If the students have previously encountered the operations of addition and subtraction of integers using a number line, then the desirability of developing the operation of multiplication in a similar setting is obvious. Many of the more common physical models that can be used to represent the product of positive and negative integers are unsuitable for interpretation on a number line. Partial success is possible in the cases of the product of two positive integers and the product of a positive and a negative integer. But, how does one illustrate that the product of two negative integers is equal to a positive integer?

REPRESENTATIONS OF ARITHMETIC PROGRESSIONS BY POSITIVE DEFINITE QUADRATIC FORMS

2011 ◽  
Vol 07 (06) ◽  
pp. 1603-1614 ◽  
Author(s):  
BYEONG-KWEON OH
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Negative Integer ◽  
Definite Integral ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms

For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

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