scholarly journals Recursions and divisibility properties for combinatorial Macdonald polynomials

2011 ◽  
Vol Vol. 13 no. 1 (Combinatorics) ◽  
Author(s):  
Nicholas A. Loehr ◽  
Elizabeth Niese
Keyword(s):  
Hilbert Series ◽  
Permutation Statistics ◽  
Macdonald Polynomial ◽  
Integer Partition ◽  
Theoretical Methods ◽  
Major Index ◽  
International Audience ◽  
Definition Of ◽  
Fermionic Formulas ◽  
Divisibility Properties

Combinatorics International audience For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.

scholarly journals Enumerating projective reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Riccardo Biagioli ◽  
Fabrizio Caselli
Keyword(s):  
Representation Theory ◽  
Generating Functions ◽  
Special Class ◽  
Hilbert Series ◽  
Weyl Groups ◽  
Reflection Groups ◽  
One Dimensional ◽  
Complex Reflection Groups ◽  
Major Index ◽  
International Audience

International audience Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated. Les groupes de réflexions projectifs ont été récemment définis par le deuxième auteur. Ils comprennent une classe spéciale de groupes notée G(r,p,s,n), qui contient tous les groupes de Weyl classiques et plus généralement tous les groupes de réflexions complexes du type G(r,p,n). Dans ce papier on définit des statistiques analogues au nombre de descentes et à l'indice majeur pour les groupes G(r,p,s,n), et on calcule plusieurs fonctions génératrices. Certains aspects de la théorie des représentations de G(r,p,s,n), comme la distribution des caractères linéaires et le calcul de la série de Hilbert de quelques algèbres d'invariants, sont aussi abordés.

scholarly journals Hall-Littlewood Polynomials in terms of Yamanouchi words

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
International Audience ◽  
Leading Term ◽  
Definition Of

International audience This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

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.

scholarly journals 0-Hecke algebra actions on coinvariants and flags

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Generating Functions ◽  
Hecke Algebra ◽  
Flag Variety ◽  
Major Index ◽  
Inversion Number ◽  
International Audience ◽  
Descent Set ◽  
Complete Flag ◽  
Coinvariant Algebra

International audience By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. En étudiant l'action de l'algèbre de 0-Hecke sur l'algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions génératrices qui comptent les permutations avec un ensemble inverse de descentes fixé, selon leur nombre d'inversions et leur "major index''.

scholarly journals Generalized Tesler matrices, virtual Hilbert series, and Macdonald polynomial operators

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Andrew Timothy Wilson
Keyword(s):  
Symmetric Function ◽  
Hilbert Series ◽  
Inner Product ◽  
Binomial Coefficients ◽  
Diagonal Harmonics ◽  
Ordered Set Partitions ◽  
International Audience ◽  
Tesler Matrices ◽  
Matrix Expression ◽  
Polynomial Operators

International audience We generalize previous definitions of Tesler matrices to allow negative matrix entries and non-positive hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices. Our interpretation uses <i>virtual Hilbert series</i>, a new class of symmetric function specializations which are defined by their values on (modified) Macdonald polynomials. As a result of this interpretation, we obtain a Tesler matrix expression for the Hall inner product $\langle \Delta_f e_n, p_{1^{n}}\rangle$, where $\Delta_f$ is a symmetric function operator from the theory of diagonal harmonics. We use our Tesler matrix expression, along with various facts about Tesler matrices, to provide simple formulas for $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ and $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ involving $q; t$-binomial coefficients and ordered set partitions, respectively. Nous généralisons les définitions précédentes de matrices Tesler pour permettre les entrées de la matrice négatives et des montants crochet non-positifs. Notre principal résultat est une interprétation algébrique d’une certaine somme pondérée sur ces matrices. Notre interprétation utilise <i>série de Hilbert virtuel</i>, une nouvelle classe de spécialisations fonctionnelles symétriques qui sont définies par leurs valeurs sur les polynômes de Macdonald (modifiées). À la suite de cette interprétation, on obtient une expression de la matrice Tesler pour la salle intérieure produit $\langle \Delta_f e_n, p_{1^{n}}\rangle$, où $\Delta_f$ est un opérateur de fonction symétrique de la théorie harmonique de diagonale. Nous utilisons notre expression de la matrice Tesler, ainsi que divers faits sur des matrices Tesler, de fournir des formules simples pour $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ et $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ impliquant $q; t$-coefficients binomial et ensemble ordonné partitions, respectivement.

scholarly journals Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

10.37236/2001 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Petter Brändén ◽  
Anders Claesson
Keyword(s):  
Fixed Points ◽  
Relative Position ◽  
Permutation Statistics ◽  
Permutation Patterns ◽  
Alternating Permutations ◽  
Major Index ◽  
Permutation Pattern ◽  
New Type ◽  
Permutation Statistic ◽  
Infinite Linear

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

scholarly journals On the minimal distance of a polynomial code

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Peter Pal Pach ◽  
Csaba Szabo
Keyword(s):  
Finite Subset ◽  
60Th Birthday ◽  
Theoretical Problem ◽  
Minimal Distance ◽  
Distribution Of Primes ◽  
Special Issue ◽  
Theoretical Methods ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Special Case

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.

scholarly journals Zonotopes, toric arrangements, and generalized Tutte polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Luca Moci
Keyword(s):  
Characteristic Polynomial ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Tutte Polynomial ◽  
Integral Points ◽  
Poincaré Polynomial ◽  
Tutte Polynomials ◽  
International Audience ◽  
Toric Arrangement ◽  
Positive Coefficients

International audience We introduce a multiplicity Tutte polynomial $M(x,y)$, which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that $M(x,y)$ satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial $M(x,y)$, likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, $M(1,y)$ is the Hilbert series of the related discrete Dahmen-Micchelli space, while $M(x,1)$ computes the volume and the number of integral points of the associated zonotope. On introduit un polynôme de Tutte avec multiplicité $M(x, y)$, qui généralise le polynôme de Tutte ordinaire et a des applications aux zonotopes et aux arrangements toriques. Nous prouvons que $M(x, y)$ satisfait une récurrence de "deletion-restriction'' et a des coefficients positifs. Le polynôme caractéristique et le polynôme de Poincaré d'un arrangement torique sont des spécialisations du polynôme associé $M(x, y)$, de même que les polynômes correspondants pour un arrangement d'hyperplans sont des spécialisations du polynôme de Tutte ordinaire. En outre, $M(1, y)$ est la série de Hilbert de l'espace discret de Dahmen-Micchelli associé, et $M(x, 1)$ calcule le volume et le nombre de points entiers du zonotope associé.

scholarly journals Crystal energy via charge

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Anne Schilling
Keyword(s):  
Energy Function ◽  
Tensor Products ◽  
Macdonald Polynomials ◽  
Recursive Definition ◽  
R Matrix ◽  
Single Column ◽  
Nous Montrons ◽  
International Audience ◽  
Definition Of

International audience The Ram–Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types ${A}$ and ${C}$ it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we show that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler than the recursive definition of energy in terms of the combinatorial ${R}$-matrix. La formule de Ram et Yip pour les polynômes de Macdonald (à t = 0) fournit une statistique que nous appelons la charge. Dans les types ${A}$ et ${C}$, elle peut être définie sur les produits tensoriels des cristaux pour les colonnes de Kashiwara–Nakashima. Dans ce papier, nous montrons que la charge est égale à (l'opposé de) la fonction d'énergie sur cristaux affines. L'algorithme pour calculer la charge est bien plus simple que la définition récursive de l'énergie en fonction de la ${R}$-matrice combinatoire.

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