scholarly journals The polytope of Tesler matrices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales ◽  
Brendon Rhoades
Keyword(s):  
Partition Function ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Integer Points ◽  
International Audience ◽  
Kostant Partition Function ◽  
Tesler Matrices ◽  
Flow Polytope ◽  
Standard Young Tableaux

26 pages, 4 figures. v2 has typos fixed, updated references, and a final remarks section including remarks from previous sections International audience We introduce the Tesler polytope $Tes_n(a)$, whose integer points are the Tesler matrices of size n with hook sums $a_1,a_2,...,a_n in Z_{\geq 0}$. We show that $Tes_n(a)$ is a flow polytope and therefore the number of Tesler matrices is counted by the type $A_n$ Kostant partition function evaluated at $(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$. We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of $Tes_n(a)$ when all $a_i>0$ is given by the Mahonian numbers and calculate the volume of $Tes_n(1,1,...,1)$ to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape. On présente le polytope de Tesler $Tes_n(a)$, dont les points réticuilaires sont les matrices de Tesler de taillen avec des sommes des équerres $a_1,a_2,...,a_n in Z_{\geq 0}$. On montre que $Tes_n(a)$ est un polytope de flux. Donc lenombre de matrices de Tesler est donné par la fonction de Kostant de type An évaluée à ($(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$On décrit les faces de ce polytope en termes de “tableaux de Tesler” et on caractérise quand le polytope est simple.On montre que l’h-vecteur de $Tes_n(a)$ , quand tous les $a_i>0$ , est donnée par le nombre de permutations avec unnombre donné d’inversions et on calcule le volume de T$Tes_n(1,1,...,1)$ comme un produit de nombres de Catalanconsécutives multiplié par le nombre de tableaux standard de Young en forme d’escalier

scholarly journals Pattern avoidance in alternating permutations and tableaux (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Alternating Permutations ◽  
Special Cases ◽  
Nous Montrons ◽  
Insertion Algorithm ◽  
Generating Trees ◽  
International Audience ◽  
Standard Young Tableaux

International audience We give bijective proofs of pattern-avoidance results for a class of permutations generalizing alternating permutations. The bijections employed include a modified form of the RSK insertion algorithm and recursive bijections based on generating trees. As special cases, we show that the sets $A_{2n}(1234)$ and $A_{2n}(2143)$ are in bijection with standard Young tableaux of shape $\langle 3^n \rangle$. Alternating permutations may be viewed as the reading words of standard Young tableaux of a certain skew shape. In the last section of the paper, we study pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape $\lambda / \mu$ whose reading words avoid $213$ is a natural $\mu$-analogue of the Catalan numbers. Similar results for the patterns $132$, $231$ and $312$. Nous présentons des preuves bijectives de résultats pour une classe de permutations à motifs exclus qui généralisent les permutations alternantes. Les bijections utilisées reposent sur une modification de l'algorithme d'insertion "RSK" et des bijections récursives basées sur des arbres de génération. Comme cas particuliers, nous montrons que les ensembles $A_{2n}(1234)$ et $A_{2n}(2143)$ sont en bijection avec les tableaux standards de Young de la forme $\langle 3^n \rangle$. Une permutation alternante peut être considérée comme le mot de lecture de certain skew tableau. Dans la dernière section de l'article, nous étudions l'évitement des motifs dans les mots de lecture de skew tableaux généraux. Nous montrons bijectivement que le nombre de tableaux standards de forme $\lambda / \mu$ dont les mots de lecture évitent $213$ est un $\mu$-analogue naturel des nombres de Catalan. Des résultats analogues sont valables pour les motifs $132$, $231$ et $312$.

scholarly journals A direct bijective proof of the hook-length formula

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Igor Pak ◽  
Alexander V. Stoyanovskii
Keyword(s):  
Young Tableaux ◽  
Hook Length ◽  
Bijective Proof ◽  
International Audience ◽  
Standard Young Tableaux

International audience This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.

scholarly journals The Robinson―Schensted Correspondence and $A_2$-webs

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Matthew Housley ◽  
Heather M. Russell ◽  
Julianna Tymoczko
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Quantum Group ◽  
Classical Limit ◽  
Young Tableaux ◽  
Tensor Power ◽  
Recent Interest ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Graphical Algorithm

International audience The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.

scholarly journals On the Poset and Asymptotics of Tesler Matrices

10.37236/6877 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Jason O'Neill
Keyword(s):  
Partition Function ◽  
Characteristic Polynomial ◽  
Parking Functions ◽  
Integral Matrices ◽  
Diagonal Harmonics ◽  
Kostant Partition Function ◽  
Tesler Matrices

Tesler matrices are certain integral matrices counted by the Kostant partition function and have appeared recently in Haglund's study of diagonal harmonics. In 2014, Drew Armstrong defined a poset on such matrices and conjectured that the characteristic polynomial of this poset is a power of $q-1$. We use a method of Hallam and Sagan to prove a stronger version of this conjecture for posets of a certain class of generalized Tesler matrices. We also study bounds for the number of Tesler matrices and how they compare to the number of parking functions, the dimension of the space of diagonal harmonics.

scholarly journals The Selberg integral and Young books

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jang Soo Kim ◽  
Suho Oh
Keyword(s):  
Young Tableaux ◽  
Combinatorial Interpretation ◽  
Selberg Integral ◽  
Combinatorial Objects ◽  
Special Cases ◽  
International Audience ◽  
Enumeration Formula ◽  
Standard Young Tableaux

International audience The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books'' are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, enumeration formulas for standard Young tableaux of these shapes are obtained. L’intégrale de Selberg est une partie intégrante importante abord évaluée par Selberg en 1944. Stanley a trouvé une interprétation combinatoire de la Selberg aide en permutations. Dans ce papier, de nouveaux objets combinatoires “livres de Young” sont introduits et présentés à avoir un lien avec l’intégrale de Selberg. Cette connexion donne une formule d'énumération pour les livres de Young. Il est démontré que des cas spéciaux de livres de Young deviennent tableaux standards de Young de formes diverses: escaliers décalés, places, certaines formes gauches et certaines formes tronquées. En conséquence, l’énumération des formules pour tableaux standards de Young de ces formes sont obtenues.

scholarly journals Kirillov's Unimodality Conjecture for the Rectangular Narayana Polynomials

10.37236/6806 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Herman Z. Q. Chen ◽  
Arthur L. B. Yang ◽  
Philip B. Zhang
Keyword(s):  
Generating Function ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Rectangular Shape ◽  
The Real ◽  
Real Zeros ◽  
Kostka Numbers ◽  
Standard Young Tableaux

In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of certain rectangular shape, up to a power of the indeterminate. Then we obtain the real-rootedness of the rectangular Narayana polynomial based on a result of Brenti which implies that the descent generating function of standard Young tableaux has only real zeros.

scholarly journals A canonical basis for Garsia-Procesi modules

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Jonah Blasiak
Keyword(s):  
Positive Integer ◽  
Canonical Basis ◽  
Hecke Algebra ◽  
Type A ◽  
Young Tableaux ◽  
Affine Hecke Algebra ◽  
International Audience ◽  
Standard Young Tableaux

International audience We identify a subalgebra $\widehat{\mathscr{H}}^+_n$ of the extended affine Hecke algebra $\widehat{\mathscr{H}}_n$ of type $A$. The subalgebra $\widehat{\mathscr{H}}^+_n$ is a u-analogue of the monoid algebra of $\mathcal{S}_n ⋉ℤ_≥0^n$ and inherits a canonical basis from that of $\widehat{\mathscr{H}}_n$. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod $n$, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient $\mathscr{R}_1^n$ of $\widehat{\mathscr{H}}^+_n$ that is a $u$-analogue of the ring of coinvariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element $*π ∈ \widehat{\mathscr{H}}^+_n$ corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that $\mathscr{R}_1^n$ has cellular quotients $\mathscr{R}_λ$ that are $u$-analogues of the Garsia-Procesi modules $R_λ$ with left cells labeled by (a PAT version of) the $λ$ -catabolizable tableaux. On définit une sous-algèbre $\widehat{\mathscr{H}}^+_n$ de l'extension affine de l'algèbre de Hecke \$\widehat{\mathscr{H}}_n$ de type $A$. La sous-algèbre $\widehat{\mathscr{H}}^+_n$ est $u$-analogue à l'algèbre monoïde de $\mathcal{S}_n ⋉ℤ_≥0^n$ et hérite d'une base canonique de $\widehat{\mathscr{H}}_n$. On montre que ses cellules gauches sont naturellement classées par des tableaux remplis d'entiers naturels ayant chacun des restes différents modulo $n$, que l'on nomme Positive Affine Tableaux (PAT). On montre ensuite qu'un sous-quotient cellulaire $\mathscr{R}_1^n$ de $\widehat{\mathscr{H}}^+_n$ est une $u$-analogue de l'anneau des co-invariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ avec des cellules gauches classées PAT qui sont essentiellement des tableaux de Young standards avec des labels cochargés. Multiplier les éléments de la base canonique par un certain élément $π ∈ \widehat{\mathscr{H}}^+_n$ correspond à des rotations de mots, et par rapport aux cellules cela correspond à un cocyclage. Plus loin, on montre que $\mathscr{R}_1^n$ a pour quotients cellulaires $\mathscr{R}_λ$ qui sont $u$- analogues aux modules de Garsia-Procesi $R_λ$ avec des cellules gauches définies par (une version PAT) des tableaux $λ$ -catabolisable.

scholarly journals Number of standard strong marked tableaux

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Susanna Fishel ◽  
Matjaž Konvalinka
Keyword(s):  
Schur Functions ◽  
Young Tableaux ◽  
Nous Proposons ◽  
Hook Length ◽  
International Audience ◽  
Standard Young Tableaux

International audience Many results involving Schur functions have analogues involving $k-$Schur functions. Standard strong marked tableaux play a role for $k-$Schur functions similar to the role standard Young tableaux play for Schur functions. We discuss results and conjectures toward an analogue of the hook length formula. De nombreux résultats impliquant les fonctions de Schur possèdent des analogues pour les fonctions de k-Schur. Les tableaux standard fortement marqués jouent un rôle pour les fonctions de k-Schur semblable á celui joué par les tableaux de Young pour les fonctions de Schur. Nous proposons ici des résultats et conjectures vers un analogue de la formule des équerres.

scholarly journals Symmetry properties of the Novelli-Pak-Stoyanovskii algorithm

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Robin Sulzgruber
Keyword(s):  
Young Tableau ◽  
Sorting Algorithm ◽  
Young Tableaux ◽  
Hook Length ◽  
Jeu De Taquin ◽  
Bijective Proof ◽  
Symmetry Properties ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

International audience The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.

scholarly journals Flow polytopes and the Kostant partition function

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales
Keyword(s):  
Partition Function ◽  
Catalan Number ◽  
Signed Graphs ◽  
Volume Formula ◽  
Flow Polytopes ◽  
International Audience ◽  
Kostant Partition Function ◽  
Mise En Pratique ◽  
The Relationship ◽  
Special Case

International audience We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As an application of our results we study a distinguished family of flow polytopes: the Chan-Robbins-Yuen polytopes. Inspired by their beautiful volume formula $\prod_{k=0}^{n-2} Cat(k)$ for the type $A_n$ case, where $Cat(k)$ is the $k^{th}$ Catalan number, we introduce type $C_{n+1}$ and $D_{n+1}$ Chan-Robbins-Yuen polytopes along with intriguing conjectures about their volumes. Nous établissons la relation entre les volumes de polytopes de flux associés aux graphes signés et la fonction de partition de Kostant. Le cas particulier de cette relation où les graphes ne sont pas signés a été étudié en détail par Baldoni et Vergne en utilisant des techniques de résidus. Contrairement à leur approche, nous apportons des preuves combinatoires inspirées par l'analyse de Postnikov et Stanley sur les polytopes de flux. Comme mise en pratique des résultats, nous étudions une famille distinguée de polytopes de flux: les polytopes Chan-Robbins-Yuen. Inspirés par leur belle formule du volume $\prod_{k=0}^{n-2} Cat(k)$ pour le cas de type $A_n$ (où $Cat(k)$ est le $k$-ème nombres de Catalan), nous présentons les polytopes Chan-Robbins-Yuen des types $C_{n +1}$ et $D_{n +1}$ accompagnés de conjectures intéressantes sur leurs volumes.

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