scholarly journals Piecewise-linear and birational toggling

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
David Einstein ◽  
James Propp
Keyword(s):  
Symmetry Property ◽  
Piecewise Linear ◽  
Young Tableaux ◽  
Fonction Transfert ◽  
International Audience ◽  
Reciprocal Symmetry ◽  
Transfer Map ◽  
Order Ideals

International audience We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset $P$ as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley's transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When $P = [a] \times [b]$, a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. We prove some homomesy results, showing that for certain functions $f$, the average of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen. Nous définissons et étudions certains analogues linéaires-par-morceaux et birationnels d’involutions toggles, rowmotion et promotion sur les idéaux d’un poset $P$, comme étudié par Striker et Williams. La rowmotion linéaire-par-morceaux est liée à la fonction transfert de Stanley pour les polytopes d’ordre; la promotion linéaire-par-morceaux se rapporte à la promotion de Schützenberger pour les tableaux semi-standards de Young. Lorsque $P = [a] \times [b]$, une propriété de symétrie réciproque récemment prouvée par Grinberg et Roby implique que la rowmotion birationnelle (et par conséquent la rowmotion linéaire-par-morceaux) est de l’ordre $a+b$. Nous démontrons quelques résultats d’homomésie, montrant que pour certaines fonctions $f$, la moyenne de $f$ sur chaque orbite de rowmotion/promotion est indépendante de l’orbite choisie.

scholarly journals The order of birational rowmotion

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Darij Grinberg ◽  
Tom Roby
Keyword(s):  
Recent Work ◽  
Finite Order ◽  
Piecewise Linear ◽  
Finite Poset ◽  
Natural Operation ◽  
International Audience ◽  
Order Ideals ◽  
Linear Setting

International audience Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting. In the latter setting, it is no longer \empha priori clear even that birational rowmotion has finite order, and for many posets the order is infinite. However, we are able to show that birational rowmotion has the same order, p+q, for the poset P=[p]×[q] (product of two chains), as ordinary rowmotion. We also show that birational (hence ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture.

scholarly journals The poset perspective on alternating sign matrices

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Generating Function ◽  
Young Tableaux ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
Combinatorial Objects ◽  
Symmetric Plane ◽  
International Audience ◽  
Order Ideals ◽  
Nous Utilisons

International audience Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

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 Boolean complexes and boolean numbers

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Bridget Eileen Tenner
Keyword(s):  
Coxeter Group ◽  
Bruhat Order ◽  
Graph Invariant ◽  
Coxeter System ◽  
Combinatorial Properties ◽  
Nous Montrons ◽  
International Audience ◽  
Order Ideals ◽  
Simplicial Poset ◽  
The Ideal

International audience The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par L'ordre de Bruhat munit tout groupe de Coxeter d'une structure de poset. L'idéal composé des éléments de ce poset engendrant des idéaux principaux ordonnés booléens, forme un poset simplicial. Ce poset simplicial définit le complexe booléen pour le groupe. Dans un système de Coxeter de rang n, nous montrons que le complexe booléen est homotopiquement équivalent à un bouquet de sphères de dimension (n-1). Le nombre de ces sphères est le nombre booléen, qui peut être calculé inductivement à partir du système de Coxeter non-étiquetté; définissant ainsi un invariant de graphe. Pour certaines familles de graphes, les nombres booléens satisfont des propriétés combinatoires intriguantes. Ce travail est une collaboration entre Claesson, Kitaev, et Ragnarsson.

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 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 The Many Faces of Alternating-Sign Matrices

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
James Propp
Keyword(s):  
Height Function ◽  
Original Form ◽  
Alternating Sign Matrices ◽  
International Audience ◽  
Order Ideals ◽  
The Many

International audience I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops.

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 Rectangular Young tableaux and the Jacobi ensemble

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Marchal
Keyword(s):  
Young Tableaux ◽  
Random Surface ◽  
Jacobi Ensemble ◽  
International Audience ◽  
Deterministic Limit

International audience It has been shown by Pittel and Romik that the random surface associated with a large rectangular Youngtableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle.We show that in the corner, these fluctuations are gaussian whereas, away from the corner and when the rectangle isa square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with theJacobi ensemble.

scholarly journals Affine type A geometric crystal structure on the Grassmannian

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Gabriel Frieden
Keyword(s):  
Crystal Structure ◽  
Type A ◽  
Young Tableaux ◽  
Rectangular Shape ◽  
International Audience ◽  
Affine Type

International audience We construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator e 0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plu ̈cker coordinates of the Grassmannian.

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