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Jang Soo Kim

Publications and source records attributed to Jang Soo Kim.

At least 55 records · Page 3Linked to original sources
arXiv2018

Product formulas for certain skew tableaux

The hook length formula gives a product formula for the number of standard Young tableaux of a partition shape. The number of standard Young tableaux of a skew shape does not always have a product formula. However, for some special skew shapes, there is a product formula. Recently, Morales, Pak and Panova joint with Krattenthaler conjectured a product formula for the number of standard Young tableaux of shape $λ/μ$ for $λ=((2a+c)^{c+a},(a+c)^a)$ and $μ=(a+1,a^{a-1},1)$. They also conjectured a product formula for the number of standard Young tableaux of a certain skew shifted shape. In this paper we prove their conjectures using Selberg-type integrals. We also give a generalization of MacMahon's box theorem and a product formula for the trace generating function for a certain skew shape, which is a generalization of a recent result of Morales, Pak and Panova.

math.CO↗
arXiv2018

Proof of Chapoton's conjecture on Newton polygons of $q$-Ehrhart polynomials

Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial of an order polytope. In this paper, we prove Chapoton's conjecture.

math.CO↗
arXiv2017

Reverse plane partitions of skew staircase shapes and $q$-Euler numbers

Recently, Naruse discovered a hook length formula for the number of standard Young tableaux of a skew shape. Morales, Pak and Panova found two $q$-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). As an application of their formula, they expressed certain $q$-Euler numbers, which are generating functions for SSYTs and RPPs of a zigzag border strip, in terms of weighted Dyck paths. They found a determinantal formula for the generating function for SSYTs of a skew staircase shape and proposed two conjectures related to RPPs of the same shape. One conjecture is a determinantal formula for the number of \emph{pleasant diagrams} in terms of Schröder paths and the other conjecture is a determinantal formula for the generating function for RPPs of a skew staircase shape in terms of $q$-Euler numbers. In this paper, we show that the results of Morales, Pak and Panova on the $q$-Euler numbers can be derived from previously known results due to Prodinger by manipulating continued fractions. These $q$-Euler numbers are naturally expressed as generating functions for alternating permutations with certain statistics involving \emph{maj}. It has been proved by Huber and Yee that these $q$-Euler numbers are generating functions for alternating permutations with certain statistics involving \emph{inv}. By modifying Foata's bijection we construct a bijection on alternating permutations which sends the statistics involving \emph{maj} to the statistic involving \emph{inv}. We also prove the aforementioned two conjectures of Morales, Pak and Panova.

math.CO↗
arXiv2017

Volumes of generalized Chan-Robbins-Yuen polytopes

The normalized volume of the Chan-Robbins-Yuen polytope ($CRY_n$) is the product of consecutive Catalan numbers. The polytope $CRY_n$ has captivated combinatorial audiences for over a decade, as there is no combinatorial proof for its volume formula. In their quest to understand $CRY_n$ better, the third author and Morales introduced two natural generalizations of it and conjectured that their volumes are certain powers of $2$ multiplied by a product of consecutive Catalan numbers. Zeilberger proved one of these conjectures. In this paper we present proofs of both conjectures.

math.CO↗
arXiv2017

Weight multiplicities and Young tableaux through affine crystals

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac--Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the {\em $($spin$)$ rigid tableaux}, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac--Moody algebras. By applying combinatorics of tableaux such as the Robinson--Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles.

math.RT↗
arXiv2016

Flow polytopes with Catalan volumes

The Chan-Robbins-Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector $(1, 0, \ldots, 0, -1)$. The normalized volume of the Chan-Robbins-Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector $(1,1, 0, \ldots, 0, -2)$. We show that the volume of this polytope is a certain power of $2$ times the product of consecutive Catalan numbers. Our proof uses constant term identities and further deepens the combinatorial mystery of why these numbers appear. In addition we introduce two more families of flow polytopes whose volumes are given by product formulas.

math.CO↗
arXiv2016

On q-integrals over order polytopes

A combinatorial study of multiple $q$-integrals is developed. This includes a $q$-volume of a convex polytope, which depends upon the order of $q$-integration. A multiple $q$-integral over an order polytope of a poset is interpreted as a generating function of linear extensions of the poset. Specific modifications of posets are shown to give predictable changes in $q$-integrals over their respective order polytopes. This method is used to combinatorially evaluate some generalized $q$-beta integrals. One such application is a combinatorial interpretation of a $q$-Selberg integral. New generating functions for generalized Gelfand-Tsetlin patterns and reverse plane partitions are established. A $q$-analogue to a well known result in Ehrhart theory is generalized using $q$-volumes and $q$-Ehrhart polynomials.

math.CO↗
arXiv2016

On the f-vectors of Gelfand-Cetlin polytopes

A Gelfand-Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GC-polytope in terms of so called the face structure of a ladder diagram. Using our description, we obtain a partial differential equation whose solution is the exponential generating function of f-vectors of GC-polytopes. This solves the open problem (2) posed by Gusev, Kritchenko, and Timorin in [GKT].

math.CO↗
arXiv2015

The Combinatorics of Associated Laguerre Polynomials

The explicit double sum for the associated Laguerre polynomials is derived combinatorially. The moments are described using certain statistics on permutations and permutation tableaux. Another derivation of the double sum is provided using only the moment generating function.

math.CO↗
arXiv2014

A new $q$-Selberg integral, Schur functions, and Young books

Recently, Kim and Oh expressed the Selberg integral in terms of the number of Young books which are a generalization of standard Young tableaux of shifted staircase shape. In this paper the generating function for Young books according to major index statistic is considered. It is shown that this generating function can be written as a Jackson integral which gives a new $q$-Selberg integral. It is also shown that the new $q$-Selberg integral has an expression in terms of Schur functions.

math.CO↗
arXiv2014

Generalized Dyck tilings

Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then we introduce two natural generalizations: $k$-Dyck tilings and symmetric Dyck tilings. We are led to consider Stirling permutations, and define an analog of the Bruhat order on them. We show that certain families of $k$-Dyck tilings are in bijection with intervals in this order. We also enumerate symmetric Dyck tilings.

math.CO↗
arXiv2014

The Selberg integral and Young books

The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books" are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, product formulas for the number of standard Young tableaux of these shapes are obtained.

math.CO↗
arXiv2014

Bootstrapping and Askey-Wilson polynomials

The mixed moments for the Askey-Wilson polynomials are found using a bootstrapping method and connection coefficients. A similar bootstrapping idea on generating functions gives a new Askey-Wilson generating function. An important special case of this hierarchy is a polynomial which satisfies a four term recurrence, and its combinatorics is studied.

math.CA↗
arXiv2013

Dyck tilings, increasing trees, descents, and inversions

Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths, in which tiles are no larger than the tiles they cover. These tilings arise in the study of certain statistical physics models and also Kazhdan--Lusztig polynomials. We give two bijections between cover-inclusive Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the second bijection maps the "discrepancy" between the upper and lower boundary of the tiling to descents of the linear extension.

math.CO↗
arXiv2013

Moments of Askey-Wilson polynomials

New formulas for the nth moment of the Askey-Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured.

math.CO↗
arXiv2012

Cyclic sieving phenomenon on annular noncrossing permutations

We show an instance of the cyclic sieving phenomenon on annular noncrossing permutations with given cycle types. We define annular $q$-Kreweras numbers, annular $q$-Narayana numbers, and annular $q$-Catalan number, all of which are polynomials in $q$. We then show that these polynomials exhibit the cyclic sieving phenomenon on annular noncrossing permutations. We also show that a sum of annular $q$-Kreweras numbers becomes an annular $q$-Narayana number and a sum of $q$-Narayana numbers becomes an annular $q$-Catalan number.

math.CO↗
arXiv2012

Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

Recently, Kenyon and Wilson introduced a certain matrix $M$ in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix $M^{-1}$ is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of $M^{-1}$. In this paper we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of $M^{-1}$ is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings.

math.CO↗