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

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 Hook-content Formulae for Symplectic and Orthogonal Tableaux

2012 ◽  
Vol 55 (3) ◽  
pp. 462-473
Author(s):  
Peter S. Campbell ◽  
Anna Stokke
Keyword(s):  
Young Diagram ◽  
Schur Functions ◽  
Schur Function ◽  
Young Tableaux

AbstractBy considering the specialisation sλ(1, q, q2, … , qn–1) of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape λ in terms of the contents and hook lengths of the boxes in the Young diagram. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal λ-tableaux.

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 Refinements of the Littlewood-Richardson rule

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
J. Haglund ◽  
K. Luoto ◽  
S. Mason ◽  
S. van Willigenburg
Keyword(s):  
Schur Functions ◽  
Schur Function ◽  
Combinatorial Formula ◽  
Demazure Characters ◽  
International Audience ◽  
Demazure Character ◽  
Quasisymmetric Schur Functions

International audience We refine the classical Littlewood-Richardson rule in several different settings. We begin with a combinatorial rule for the product of a Demazure atom and a Schur function. Building on this, we also describe the product of a quasisymmetric Schur function and a Schur function as a positive sum of quasisymmetric Schur functions. Finally, we provide a combinatorial formula for the product of a Demazure character and a Schur function as a positive sum of Demazure characters. This last rule implies the classical Littlewood-Richardson rule for the multiplication of two Schur functions. Nous décrivons trois nouvelles règles de Littlewood-Richardson, et chaque nouvelle règle partage la vieille règle de Littlewood-Richardson. La première règle multiplie un atome de Demazure et une fonction de Schur. La deuxième multiplie une fonction de quasisymmetric-Schur et une fonction de Schur. La troisième multiplie un caractère de Demazure et une fonction de Schur. Cette dernière règle est une description de la vieille règle de Littlewood-Richardson.

scholarly journals On Pattern-Avoiding Partitions

10.37236/763 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Stirling Numbers ◽  
Equivalence Classes ◽  
Catalan Numbers ◽  
Set Partition ◽  
Combinatorial Interpretations

A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$. We choose the ordering of the blocks so that they satisfy $\min B_1 < \min B_2 < \cdots < \min B_d$. We represent such a set partition by a canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, with $\pi_i=j$ if $i\in B_j$. We say that a partition $\pi$ contains a partition $\sigma$ if the canonical sequence of $\pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $\sigma$. Two partitions $\sigma$ and $\sigma'$ are equivalent, if there is a size-preserving bijection between $\sigma$-avoiding and $\sigma'$-avoiding partitions. We determine all the equivalence classes of partitions of size at most $7$. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most $3$. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

scholarly journals Tokuyama's Identity for Factorial Schur $P$ and $Q$ Functions

10.37236/4971 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Partition Function ◽  
Lattice Paths ◽  
Schur Function ◽  
Vertex Model ◽  
Shifted Tableaux

A recent paper of Bump, McNamara and Nakasuji introduced a factorial version of Tokuyama's identity, expressing the partition function of  six vertex model as the product of a $t$-deformed Vandermonde and a Schur function. Here we provide an extension of their result by exploiting the language of primed shifted tableaux, with its proof based on the use of non-interesecting lattice paths.

scholarly journals Bracing Zonohedra With Special Faces

2015 ◽  
Vol 3 (1-2) ◽  
pp. 88-95 ◽  
Author(s):  
Gyula Nagy
Keyword(s):  
Convex Polyhedron ◽  
Three Dimensional ◽  
Equivalence Classes ◽  
Frame Structure ◽  
Sufficient Condition ◽  
Preliminary Design ◽  
Necessary And Sufficient Condition ◽  
Special Assumption ◽  
Centrally Symmetric ◽  
Necessary And Sufficient

Abstract The analysis of simpler preliminary design gives useful input for more complicated three-dimensional building frame structure. A zonohedron, as a preliminary structure of design, is a convex polyhedron for which each face possesses central symmetry. We considered zonohedron as a special framework with the special assumption that the polygonal faces can be deformed in such a way that faces remain planar and centrally symmetric, moreover the length of all edges remains unchanged. Introducing some diagonal braces we got a new mechanism. This paper deals with the flexibility of this kind of mechanisms, and investigates the rigidity of the braced framework. The flexibility of the framework can be characterized by some vectors, which represent equivalence classes of the edges. A necessary and sufficient condition for the rigidity of the braced rhombic face zonohedra is posed. A real mechanical construction, based on two simple elements, provides a CAD prototype of these new mechanisms.

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

scholarly journals An Orthosymplectic Pieri Rule

10.37236/7387 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Anna Stokke
Keyword(s):  
Linear Combination ◽  
Schur Functions ◽  
Symmetric Polynomial ◽  
Schur Function ◽  
Product Rule ◽  
Integer Coefficients ◽  
Pieri Formula ◽  
Symplectic Characters ◽  
Homogeneous Symmetric Polynomial

The classical Pieri formula gives a combinatorial rule for decomposing the product of a Schur function and a complete homogeneous symmetric polynomial as a linear combination of Schur functions with integer coefficients. We give a Pieri rule for describing the product of an orthosymplectic character and an orthosymplectic character arising from a one-row partition. We establish that the orthosymplectic Pieri rule coincides with Sundaram's Pieri rule for symplectic characters and that orthosymplectic characters and symplectic characters obey the same product rule. 

A Combinatorial Interpretation Of The Wreath Product Of Schur Functions

1976 ◽  
Vol 28 (4) ◽  
pp. 879-884 ◽  
Author(s):  
Glânffrwd P. Thomas
Keyword(s):  
Wreath Product ◽  
Schur Functions ◽  
Young Tableaux ◽  
Combinatorial Interpretation

A combinatorial interpretation of Schur functions in terms of Young tableaux is well-known. (For example, see Littlewood [1] or Thomas [4]). The purpose of this paper is to present a combinatorial interpretation of the wreath product (or plethysm) of two Schur functions.

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