scholarly journals A Cornucopia of Quasi-Yamanouchi Tableaux

10.37236/8082 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
George Wang
Keyword(s):  
Schur Functions ◽  
Product Formula ◽  
Lattice Paths ◽  
Young Tableaux ◽  
Jack Polynomials ◽  
Symmetric Polynomials ◽  
Descent Set ◽  
Standard Young Tableaux ◽  
Weighted Lattice Paths ◽  
Coinvariant Algebra

Quasi-Yamanouchi tableaux are a subset of semistandard Young tableaux and refine standard Young tableaux. They are closely tied to the descent set of standard Young tableaux and were introduced by Assaf and Searles to tighten Gessel's fundamental quasisymmetric expansion of Schur functions. The descent set and descent statistic of standard Young tableaux repeatedly prove themselves useful to consider, and as a result, quasi-Yamanouchi tableaux make appearances in many ways outside of their original purpose. Some examples, which we present in this paper, include the Schur expansion of Jack polynomials, the decomposition of Foulkes characters, and the bigraded Frobenius image of the coinvariant algebra. While it would be nice to have a product formula enumeration of quasi-Yamanouchi tableaux in the way that semistandard and standard Young tableaux do, it has previously been shown by the author that there is little hope on that front. The goal of this paper is to address a handful of the numerous alternative enumerative approaches. In particular, we present enumerations of quasi-Yamanouchi tableaux using $q$-hit numbers, semistandard Young tableaux, weighted lattice paths, and symmetric polynomials, as well as the fundamental quasisymmetric and monomial quasisymmetric expansions of their Schur generating function.

scholarly journals Hook formulas for skew shapes

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Alejandro H. Morales ◽  
Igor Pak ◽  
Greta Panova
Keyword(s):  
General Formula ◽  
Schur Functions ◽  
Product Formula ◽  
Young Tableaux ◽  
Hook Length ◽  
International Audience ◽  
Border Strip ◽  
Standard Young Tableaux ◽  
Skew Schur Functions ◽  
Reverse Plane

International audience The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.

scholarly journals Singular Nonsymmetric Jack Polynomials for Some Rectangular Tableaux

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 630
Author(s):  
Charles F. Dunkl
Keyword(s):  
Tensor Product ◽  
Symmetric Groups ◽  
Young Tableaux ◽  
Jack Polynomials ◽  
Main Ingredient ◽  
Dunkl Operators ◽  
Rational Cherednik Algebra ◽  
Standard Young Tableaux ◽  
Vector Valued

In the intersection of the theories of nonsymmetric Jack polynomials in N variables and representations of the symmetric groups S N one finds the singular polynomials. For certain values of the parameter κ there are Jack polynomials which span an irreducible S N -module and are annihilated by the Dunkl operators. The S N -module is labeled by a partition of N, called the isotype of the polynomials. In this paper the Jack polynomials are of the vector-valued type, i.e., elements of the tensor product of the scalar polynomials with the span of reverse standard Young tableaux of the shape of a fixed partition of N. In particular, this partition is of shape m , m , … , m with 2 k components and the constructed singular polynomials are of isotype m k , m k for the parameter κ = 1 / m + 2 . This paper contains the necessary background on nonsymmetric Jack polynomials and representation theory and explains the role of Jucys–Murphy elements in the construction. The main ingredient is the proof of uniqueness of certain spectral vectors, namely the list of eigenvalues of the Jack polynomials for the Cherednik–Dunkl operators, when specialized to κ = 1 / m + 2 . The paper finishes with a discussion of associated maps of modules of the rational Cherednik algebra and an example illustrating the difficulty of finding singular polynomials for arbitrary partitions.

scholarly journals On Cyclic Schur-Positive Sets of Permutations

10.37236/8974 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde ◽  
Yuval Roichman
Keyword(s):  
Conjugacy Classes ◽  
Young Tableaux ◽  
Classical Notion ◽  
Descent Set ◽  
Standard Young Tableaux ◽  
Schur Positive

We introduce a notion of cyclic Schur-positivity for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schur-positive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes.  In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. The proof unveils a new equidistribution phenomenon of descent sets on permutations, provides affirmative solutions to conjectures by the last two authors and by Adin–Gessel–Reiner–Roichman, and yields new examples of Schur-positive sets.

scholarly journals A Sundaram type Bijection for SO(3): Vacillating Tableaux and Pairs of Standard Young Tableaux and Orthogonal Littlewood-Richardson Tableaux

10.37236/7713 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Judith Jagenteufel
Keyword(s):  
Orthogonal Group ◽  
Young Tableau ◽  
Young Tableaux ◽  
Tensor Power ◽  
Direct Sum ◽  
Special Orthogonal Group ◽  
Direct Sum Decomposition ◽  
Descent Set ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

Motivated by the direct-sum-decomposition of the $r^{\text{th}}$ tensor power of the defining representation of the special orthogonal group $\mathrm{SO}(2k + 1)$, we present a bijection between vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau for $\mathrm{SO}(3)$.Our bijection preserves a suitably defined descent set. Using it we determine the quasi-symmetric expansion of the Frobenius characters of the isotypic components.On the combinatorial side we obtain a bijection between Riordan paths and standard Young tableaux with 3 rows, all of even length or all of odd length.

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 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 Asymptotics for minimal overlapping patterns for generalized Euler permutations, standard tableaux of rectangular shape, and column strict arrays

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Ran Pan ◽  
Jeffrey B. Remmel
Keyword(s):  
Computer Science ◽  
Discrete Math ◽  
Theoretical Computer Science ◽  
Young Tableaux ◽  
Theoretical Computer ◽  
Rectangular Shape ◽  
Set Of Permutations ◽  
Natural Analogues ◽  
Descent Set ◽  
Standard Young Tableaux

A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal overlapping patterns in $S_j$ is at least $3 -e$. Given a permutation $\sigma$, we let $\text{Des}(\sigma)$ denote the set of descents of $\sigma$. We study the class of permutations $\sigma \in S_{kn}$ whose descent set is contained in the set $\{k,2k, \ldots (n-1)k\}$. For example, up-down permutations in $S_{2n}$ are the set of permutations whose descent equal $\sigma$ such that $\text{Des}(\sigma) = \{2,4, \ldots, 2n-2\}$. There are natural analogues of the minimal overlapping permutations for such classes of permutations and we study the proportion of minimal overlapping patterns for each such class. We show that the proportion of minimal overlapping permutations in such classes approaches $1$ as $k$ goes to infinity. We also study the proportion of minimal overlapping patterns in standard Young tableaux of shape $(n^k)$. Comment: Accepted by Discrete Math and Theoretical Computer Science. Thank referees' for their suggestions

scholarly journals On Cyclic Descents for Tableaux

2018 ◽  
Vol 2020 (24) ◽  
pp. 10231-10276 ◽  
Author(s):  
Ron M Adin ◽  
Victor Reiner ◽  
Yuval Roichman
Keyword(s):  
General Context ◽  
Young Tableaux ◽  
Schur Polynomials ◽  
Natural Notion ◽  
Descent Set ◽  
New Interpretation ◽  
Almost All ◽  
Standard Young Tableaux

Abstract The notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical. Cellini introduced a natural notion of cyclic descent set for permutations, and Rhoades introduced such a notion for SYT—but only for rectangular shapes. In this work we define cyclic extensions of descent sets in a general context and prove existence and essential uniqueness for SYT of almost all shapes. The proof applies nonnegativity properties of Postnikov’s toric Schur polynomials, providing a new interpretation of certain Gromov–Witten invariants.

scholarly journals Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants

2021 ◽  
Vol vol. 23 no. 1 (Combinatorics) ◽  
Author(s):  
Robert A. Proctor ◽  
Matthew J. Willis
Keyword(s):  
Schur Functions ◽  
Equivalence Classes ◽  
Catalan Numbers ◽  
Lattice Paths ◽  
Schur Function ◽  
Young Tableaux ◽  
Meta Data ◽  
Key Polynomials ◽  
Demazure Characters ◽  
Necessary And Sufficient

Let $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$ and $\beta$ has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape $\lambda$ whose values are row-wise bounded by the entries of $\beta$. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by $\lambda$ and $\beta$; this could be done since the pair $(\lambda, \beta)$ satisfied their "nonpermutable" condition for the sequence of terminals of an $n$-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that $\beta$ be weakly increasing. Then for each $\lambda$ we give a condition on the entries of $\beta$ for the pair $(\lambda, \beta)$ to be nonpermutable that is both necessary and sufficient. When the parts of $\lambda$ are not distinct there will be multiple row bound $n$-tuples $\beta$ that will produce the same set of tableaux. We accordingly group the bounding $\beta$ into equivalence classes and identify the most efficient $\beta$ in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by $n$ and $\lambda$ are enumerated by the number of these efficient $n$-tuples. We called these counts "parabolic Catalan numbers". It is noted that the $GL(n)$ Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants. Comment: 22 pages, 5 figures, 4 tables. Identical to v.5, except for the insertion of a reference and the DMTCS journal's publication meta data

Standard Young tableaux in a (2,1)-hook and Motzkin paths

2021 ◽  
Vol 344 (7) ◽  
pp. 112395
Author(s):  
Rosena R.X. Du ◽  
Jingni Yu
Keyword(s):  
Young Tableaux ◽  
Standard Young Tableaux ◽  
Motzkin Paths
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