scholarly journals q-Rook placements and Jordan forms of upper-triangular nilpotent matrices

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Martha Yip
Keyword(s):  
Finite Field ◽  
Weighted Sum ◽  
Canonical Forms ◽  
Combinatorial Formula ◽  
Jordan Type ◽  
Rook Placements ◽  
Nilpotent Matrices ◽  
International Audience ◽  
Jordan Forms

International audience The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $F_q$ has Jordan canonical forms indexed by partitions $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$ F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements.

scholarly journals Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices

10.37236/6888 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Martha Yip
Keyword(s):  
Finite Field ◽  
Weighted Sum ◽  
Young Tableaux ◽  
Canonical Forms ◽  
Combinatorial Formula ◽  
Rook Placements ◽  
Nilpotent Matrices ◽  
Jordan Forms ◽  
Standard Young Tableaux ◽  
Young’S Lattice

The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $\mathbb{F}_q$ has Jordan canonical forms indexed by partitions $\lambda \vdash n$. We present a combinatorial formula for computing the number $F_\lambda(q)$ of matrices of Jordan type $\lambda$ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for $F_\lambda(q)$.

scholarly journals Combinatorial Formula for the Hilbert Series of bigraded $S_n$-modules

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Meesue Yoo
Keyword(s):  
Hilbert Series ◽  
Macdonald Polynomials ◽  
Weighted Sum ◽  
Young Tableaux ◽  
Combinatorial Formula ◽  
Nous Proposons ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Littlewood Polynomial ◽  
The Way

International audience We introduce a combinatorial way of calculating the Hilbert series of bigraded $S_n$-modules as a weighted sum over standard Young tableaux in the hook shape case. This method is based on Macdonald formula for Hall-Littlewood polynomial and extends the result of $A$. Garsia and $C$. Procesi for the Hilbert series when $q=0$. Moreover, we give the way of associating the fillings giving the monomial terms of Macdonald polynomials to the standard Young tableaux. Nous introduisons une méthode combinatoire pour calculer la série de Hilbert de modules bigradués de $S_n$ comme une somme pondérée sur les tableaux de Young standards à la forme crochet. Cette méthode se fonde sur la formule Macdonald pour les polynômes Hall-Littlewood et généralise un résultat de $A$. Garsia et $C$. Procesi pour la série de Hilbert dans le cas $q=0$. De plus, nous proposons une méthode pour associer aux tableaux de Young standards les remplissages des monômes des polynômes de Macdonald.

scholarly journals Infinite Families of 2-Designs from a Class of Linear Codes Related to Dembowski-Ostrom Functions

2021 ◽  
pp. 1-15
Author(s):  
Rong Wang ◽  
Xiaoni Du ◽  
Cuiling Fan ◽  
Zhihua Niu
Keyword(s):  
Finite Field ◽  
Weight Distribution ◽  
Coding Theory ◽  
Exponential Sums ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Research Interest ◽  
Combinatorial Formula ◽  
Text Design ◽  
Fixed Weight

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial [Formula: see text]-designs have attracted lots of research interest for decades. The interplay between coding theory and [Formula: see text]-designs started many years ago. It is generally known that [Formula: see text]-designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight in a code also may hold a [Formula: see text]-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of [Formula: see text]-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of [Formula: see text]-designs are calculated explicitly.

scholarly journals Peak algebras, paths in the Bruhat graph and Kazhdan-Lusztig polynomials

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Francesco Brenti ◽  
Fabrizio Caselli
Keyword(s):  
Combinatorial Formula ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Bruhat Graph ◽  
Lusztig Polynomials

International audience We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions. On montre une formule combinatoire pour les polynômes de Kazhdan-Lusztig qui est valable en toute généralité. Cette formule est plus simple et plus explicite que toutes les autres formules connues; de plus, elle ne peut pas être simplifiée linéairement. La preuve utilise une nouvelle base pour la sous-algèbre des sommets de l’algèbre des fonctions quasi-symmetriques.

scholarly journals Counting Quiver Representations over Finite Fields Via Graph Enumeration

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Geir Helleloid ◽  
Fernando Rodriguez-Villegas
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Quiver Representations ◽  
Dimension Vector ◽  
Compte Rendu ◽  
Isomorphism Classes ◽  
International Audience ◽  
Derivatives Of ◽  
Nous Utilisons ◽  
Complete Account

International audience Let $\Gamma$ be a quiver on $n$ vertices $v_1, v_2, \ldots , v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua gave a formula for $A_{\Gamma}(\boldsymbol{\alpha}, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\mathbb{F}_q$ with dimension vector $\boldsymbol{\alpha}$. We use Hua's formula to show that the derivatives of $A_{\Gamma}(\boldsymbol{\alpha}, q)$ with respect to $q$, when evaluated at $q = 1$, are polynomials in the variables $g_{ij}$, and we can compute the highest degree terms in these polynomials. The formulas for these coefficients depend on the enumeration of certain families of connected graphs. This note simply gives an overview of these results; a complete account of this research is available on the arXiv and has been submitted for publication. Soit $\Gamma$ un carquois sur $n$ sommets $ v_1, v_2, \ldots , v_n$ avec $g_{ij}$ arêtes entre $v_i$ et $v_j$, et soit $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua a donné une formule pour $A_{\Gamma}(\boldsymbol{\alpha}, q)$, le nombre de classes d'isomorphisme absolument indécomposables de représentations de $\Gamma$ sur le corps fini $\mathbb{F}_q$ avec vecteur de dimension $\boldsymbol{\alpha}$. Nous utilisons la formule de Hua pour montrer que les dérivées de $A_{\Gamma}(\boldsymbol{\alpha}, q)$ par rapport à $q$, alors évaluée à $q=1$, sont des polynômes dans les variables $g_{ij}$, et on peut calculer les termes de plus haut degré de ces polynômes. Les formules pour ces coefficients dépendent de l'énumération de certaines familles de graphes connectés. Cette note donne simplement un aperçu de ces résultats, un compte rendu complet de cette recherche est disponible sur arXiv et a été soumis pour publication.

scholarly journals Automaticity of primitive words and irreducible polynomials

2013 ◽  
Vol Vol. 15 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Anne Lacroix ◽  
Narad Rampersad
Keyword(s):  
Finite Field ◽  
Finite Automaton ◽  
Prime Numbers ◽  
Deterministic Finite Automaton ◽  
Irreducible Polynomials ◽  
Letter Alphabet ◽  
International Audience ◽  
Primitive Words

Automata, Logic and Semantics International audience If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.

scholarly journals Dual combinatorics of zonal polynomials

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Piotr Sniady
Keyword(s):  
Irreducible Character ◽  
Symmetric Functions ◽  
Symmetric Groups ◽  
Combinatorial Formula ◽  
Zonal Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Recent Developments ◽  
International Audience ◽  
Special Case

International audience In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials. Dans cet article, nous établissons une nouvelle formule combinatoire pour les polynômes zonaux en fonction des fonctions puissance. La preuve utilise le principe de l'involution changeant les signes. Nous en déduisons des formules pour les caractères zonaux, qui sont définis comme les coefficients des polynômes zonaux écrits sur la base des fonctions puissance, normalisés de manière appropriée. Ces formules sont des analogues de développements récents sur les caractères du groupe symétrique. L'existence de telles formules aurait pu être prédite à partir des travaux de M. Lassalle, qui a proposé deux conjectures de positivité sur les caractères de Jack, que nous prouvons dans le cas particulier des polynômes zonaux.

scholarly journals On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

scholarly journals The ABC's of affine Grassmannians and Hall-Littlewood polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Avinash J. Dalal ◽  
Jennifer Morse
Keyword(s):  
Schur Functions ◽  
Type A ◽  
Combinatorial Formula ◽  
Transition Matrices ◽  
Nous Obtenons ◽  
Affine Grassmannians ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Schubert Classes ◽  
Affine Type

International audience We give a new description of the Pieri rule for $k$-Schur functions using the Bruhat order on the affine type-$A$ Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and $k$-Schur functions. Nous présentons une nouvelle description, issue de l'ordre de Bruhat du groupe de Weyl affine de type $A$, de la règle de Pieri pour les fonctions $k$-Schur. Ce faisant, nous obtenons une nouvelle formule combinatoire pour les représentants des classes de Schubert de la cohomologie des Grassmannienne affines. Nous décrivons aussi comment notre approche permet d'obtenir les polynômes de Kostka-Foulkes et comment elle peut être appliquée à l’étude des matrices de transition entre les polynômes de Hall-Littlewood et les fonctions $k$-Schur.

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