scholarly journals A Bijective Proof of the Hook-content Formula

10.37236/1272 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Christian Krattenthaler
Keyword(s):  
Schur Functions ◽  
Product Formula ◽  
Invariance Property ◽  
Jeu De Taquin ◽  
Combinatorial Description ◽  
Bijective Proof ◽  
Common Extension ◽  
The Relationship ◽  
Principal Specialization

A bijective proof of the product formula for the principal specialization of super Schur functions (also called hook Schur functions) is given using the combinatorial description of super Schur functions in terms of certain tableaux due to Berele and Regev. Our bijective proof is based on the Hillman–Grassl algorithm and a modified version of Schützenberger's jeu de taquin. We then explore the relationship between our modified jeu de taquin and a modified jeu de taquin by Goulden and Greene. We define a common extension and prove an invariance property for it, thus discovering that both modified jeu de taquins are, though different, equivalent.

scholarly journals Combinatorial Expansions in $K$-Theoretic Bases

10.37236/2320 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Geometric Information ◽  
Combinatorial Description ◽  
Littlewood Polynomials ◽  
Stanley Symmetric Functions ◽  
K Theory ◽  
Combinatorial Representation

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape.  Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space.  In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

scholarly journals A bijective proof of Macdonald's reduced word formula

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Alexander Holroyd ◽  
Benjamin Young
Keyword(s):  
Reduced Word ◽  
Bijective Proof ◽  
International Audience ◽  
Principal Specialization

International audience We describe a bijective proof of Macdonald's reduced word identity using pipe dreams and Little's bumping algorithm. The proof extends to a principal specialization of the identity due to Fomin and Stanley. Our bijective tools also allow us to address a problem posed by Fomin and Kirillov from 1997, using work of Wachs, Lenart and Serrano- Stump.

scholarly journals A 'Nice' Bijection for a Content Formula for Skew Semistandard Young Tableaux

10.37236/1635 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Martin Rubey
Keyword(s):  
Generating Function ◽  
Young Tableaux ◽  
Jeu De Taquin ◽  
Enumerative Combinatorics ◽  
Bijective Proof

Based on Schützenberger's evacuation and a modification of jeu de taquin, we give a bijective proof of an identity connecting the generating function of reverse semistandard Young tableaux with bounded entries with the generating function of all semistandard Young tableaux. This solves Exercise 7.102 b of Richard Stanley's book 'Enumerative Combinatorics 2'.

scholarly journals Jeu de taquin, uniqueness of rectification and ultradiscrete KP

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Shinsuke Iwao
Keyword(s):  
Time Evolution ◽  
Integrable System ◽  
Combinatorial Problem ◽  
Young Tableaux ◽  
Kp Equation ◽  
Jeu De Taquin ◽  
The Relationship

Abstract In this article, we study tropical-theoretic aspects of the ‘rectification algorithm’ on skew Young tableaux. It is shown that the algorithm is interpreted as a time evolution of some tropical integrable system. By using this fact, we construct a new combinatorial map that is essentially equivalent to the rectification algorithm. Some of properties of the rectification can be seen more clearly via this map. For example, the uniqueness of a rectification boils down to an easy combinatorial problem. Our method is mainly based on the two previous researches: the theory of geometric tableaux by Noumi–Yamada, and the study on the relationship between jeu de taquin slides and the ultradiscrete KP equation by Mikami and Katayama–Kakei.

scholarly journals The Relationship Between the Activity of Scouting and Learning Outcomes of Students at SMPN 40 Padang

2020 ◽  
Vol 8 (4) ◽  
pp. 499
Author(s):  
Abdi Amulya ◽  
Irmawita Irmawita
Keyword(s):  
Learning Outcomes ◽  
Confidence Level ◽  
Product Formula ◽  
Learning Achievement ◽  
Significance Level ◽  
Level Of Activity ◽  
The Moment ◽  
Relationship Of ◽  
The Relationship ◽  
Low Activity

The background of this study is about the activity of scouting the students SMPN 40 Padang is a low category and very influential on the learning achievement of students obtained. The purpose of this research is to see the levels and relationship of scouting organizational effectiveness and learning outcomes of students SMPN 40 Padang.  The type of this research is correlational with a quantitative approach. The sample of this study consisted of 30 students in SMPN 40 Padang by using random sampling data collection. The instrument of this study used a questionnaire about scouting organizational activeness consisting of 17 questions and data about the learning achievement of students of SMPN 40 Padang. From this research, it can be concluded that the level of activity in the scouting organization with the learning achievement of students at SMPN 40 Padang is categorized as low, because it can be seen from the results of the questionnaire that students responded (48.50%) rarely and never (25.40%). Meanwhile, seen from the learning outcomes of students (36.7%) in the low category and very low (10%). This is evidenced by the calculation of the correlation using the moment product formula which shows the results that recount (0.548) is greater than rtable at the 95% confidence level (0.361) and rtabel for the 99% confidence level (0.463) so that the researchers get quite significant results. The relationship is quite significant with a significance level of 99% so that the low activity of scouting organizations followed by students can also affect the learning achievement of students at SMPN 40 Padang.Keywords: Scouting organizational activeness, learning outcomes

scholarly journals Families of Major Index Distributions: Closed Forms and Unimodality

10.37236/8585 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
William J. Keith
Keyword(s):  
Schur Functions ◽  
Positive Answer ◽  
Schur Function ◽  
Young Tableaux ◽  
Large Collection ◽  
Major Index ◽  
The Family ◽  
Standard Young Tableaux ◽  
Principal Specialization

Closed forms for $f_{\lambda,i} (q) := \sum_{\tau \in SYT(\lambda) : des(\tau) = i} q^{maj(\tau)}$, the distribution of the major index over standard Young tableaux of given shapes and specified number of descents, are established for a large collection of $\lambda$ and $i$. Of particular interest is the family that gives a positive answer to a question of Sagan and collaborators. All formulas established in the paper are unimodal, most by a result of Kirillov and Reshetikhin. Many can be identified as specializations of Schur functions via the Jacobi-Trudi identities. If the number of arguments is sufficiently large, it is shown that any finite principal specialization of any Schur function $s_\lambda(1,q,q^2,\dots,q^{n-1})$ has a combinatorial realization as the distribution of the major index over a given set of tableaux.

scholarly journals Row-strict quasisymmetric Schur functions

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sarah K Mason ◽  
Jeffrey Remmel
Keyword(s):  
Schur Functions ◽  
Schur Function ◽  
Nous Obtenons ◽  
Schur Polynomials ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Quasisymmetric Schur Functions ◽  
The Relationship

International audience Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the $\textit{quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the $\textit{row-strict quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships. Haglund, Luoto, Mason, et van Willigenburg ont introduit une base pour les fonctions quasi-symétriques appelée $\textit{base des fonctions de Schur quasi-symétriques}$, qui sont construites en remplissant des diagrammes de compositions, d'une manière très semblable à la construction des fonctions de Schur à partir des tableaux "column-strict'' (ordre strict sur les colonnes). Nous introduisons une nouvelle base pour les fonctions quasi-symétriques appelée $\textit{base des fonctions de Schur quasi-symétriques "row-strict''}$, qui sont construites en remplissant des diagrammes de compositions, d'une manière très semblable à la construction des fonctions de Schur à partir des tableaux "row-strict'' (ordre strict sur les lignes). Nous décrivons la relation entre cette nouvelle base et d'autres bases connues pour les fonctions quasi-symétriques, ainsi que ses relations avec les polynômes de Schur. Nous obtenons un raffinement de l'opérateur oméga comme conséquence de ces relations.

scholarly journals The knot Floer cube of resolutions and the composition product

2020 ◽  
Vol 29 (03) ◽  
pp. 2050006
Author(s):  
Nathan Dowlin
Keyword(s):  
Euler Characteristic ◽  
Floer Homology ◽  
Product Formula ◽  
Knot Floer Homology ◽  
Direct Sum ◽  
Heegaard Diagram ◽  
Composition Product ◽  
The Relationship

We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot [Formula: see text], we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of [Formula: see text].

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 Matrices connected with Brauer's centralizer algebras

10.37236/1217 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Mark D. McKerihan
Keyword(s):  
Algebra Structure ◽  
Jeu De Taquin ◽  
Combinatorial Description ◽  
Centralizer Algebras ◽  
Centralizer Algebra

In a 1989 paper, Hanlon and Wales showed that the algebra structure of the Brauer Centralizer Algebra $A_f^{(x)}$ is completely determined by the ranks of certain combinatorially defined square matrices $Z^{\lambda / \mu}$, whose entries are polynomials in the parameter $x$. We consider a set of matrices $M^{\lambda / \mu}$ found by Jockusch that have a similar combinatorial description. These new matrices can be obtained from the original matrices by extracting the terms that are of "highest degree" in a certain sense. Furthermore, the $M^{\lambda / \mu}$ have analogues ${\cal M}^{\lambda / \mu}$ that play the same role that the $Z^{\lambda / \mu}$ play in $A_f^{(x)}$, for another algebra that arises naturally in this context. We find very simple formulas for the determinants of the matrices $M^{\lambda/\mu}$ and ${\cal M}^{\lambda / \mu}$, which prove Jockusch's original conjecture that $\det M^{\lambda / \mu}$ has only integer roots. We define a Jeu de Taquin algorithm for standard matchings, and compare this algorithm to the usual Jeu de Taquin algorithm defined by Schützenberger for standard tableaux. The formulas for the determinants of $M^{\lambda/\mu}$ and ${\cal M}^{\lambda / \mu}$ have elegant statements in terms of this new Jeu de Taquin algorithm.

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