scholarly journals Determinants of Super-Schur Functions, Lattice Paths, and Dotted Plane Partitions

1993 ◽  
Vol 98 (1) ◽  
pp. 27-64 ◽  
Author(s):  
F. Brenti
Keyword(s):  
Schur Functions ◽  
Lattice Paths ◽  
Plane Partitions

scholarly journals Signed Lozenge Tilings

10.37236/5579 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
D. Cook II ◽  
Uwe Nagel
Keyword(s):  
Perfect Matchings ◽  
Lattice Paths ◽  
New Method ◽  
Plane Partitions ◽  
Triangular Region ◽  
Lozenge Tilings

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations.

scholarly journals Tableaux and plane partitions of truncated shapes (extended abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Greta Panova
Keyword(s):  
Generating Function ◽  
Schur Functions ◽  
Young Tableaux ◽  
Young Diagrams ◽  
Plane Partitions ◽  
Nous Trouvons ◽  
International Audience

International audience We consider a new kind of straight and shifted plane partitions/Young tableaux — ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function. Nous considérons un nouveau type de partitions planes, ou de tableaux de Young, droits ou décalés, obtenus en privant leurs diagrammes de certaines cellules en haut à droite, et dans certains cas nous trouvons des formules d'énumération pour les tableaux standard. Les preuves impliquent le calcul de la fonction génératrice pour les partitions planes correspondantes, en utilisant des interprétations et des formules pour les sommes de fonctions de Schur restreintes et leurs spécialisations. Le nombre de tableaux standard est alors obtenu comme une certaine limite de cette fonction.

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 Bijections between Lattice Paths and Plane Partitions

2011 ◽  
Vol 01 (03) ◽  
pp. 108-115 ◽  
Author(s):  
Mateus Alegri ◽  
Eduardo Henrique de Mattos Brietzke ◽  
José Plínio de Oliveira Santos ◽  
Robson da Silva
Keyword(s):  
Lattice Paths ◽  
Plane Partitions

scholarly journals A triple product formula for plane partitions derived from biorthogonal polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Product Formula ◽  
Triple Product ◽  
Lattice Paths ◽  
Plane Partitions ◽  
International Audience ◽  
Biorthogonal Polynomials ◽  
Bounded Size

International audience A new triple product formulae for plane partitions with bounded size of parts is derived from a combinato- rial interpretation of biorthogonal polynomials in terms of lattice paths. Biorthogonal polynomials which generalize the little q-Laguerre polynomials are introduced to derive a new triple product formula which recovers the classical generating function in a triple product by MacMahon and generalizes the trace-type generating functions in double products by Stanley and Gansner.

scholarly journals Refined Dual Stable Grothendieck Polynomials and Generalized Bender-Knuth Involutions

10.37236/5737 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Pavel Galashin ◽  
Darij Grinberg ◽  
Gaku Liu
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
The Other ◽  
Countable Family ◽  
Plane Partitions ◽  
K Theory ◽  
Reverse Plane ◽  
Grothendieck Polynomials

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the $K$-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries $1$ and $2$.

scholarly journals Linear Recurrences for Cylindrical Networks

2017 ◽  
Vol 2019 (13) ◽  
pp. 4047-4080
Author(s):  
Pavel Galashin ◽  
Pavlo Pylyavskyy
Keyword(s):  
General Theorem ◽  
Schur Functions ◽  
Linear Recurrence ◽  
Linear Recurrences ◽  
Domino Tilings ◽  
Plane Partitions

Abstract We prove a general theorem that gives a linear recurrence for tuples of paths in every cylindrical network. This can be seen as a cylindrical analog of the Lindström–Gessel–Viennot theorem. We illustrate the result by applying it to Schur functions, plane partitions, and domino tilings.

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