A bijective proof of the generating function for the number of reverse plane partitions via lattice paths

1984 ◽  
Vol 16 (1-4) ◽  
pp. 75-91 ◽  
Author(s):  
Jeffrey B. Remmel ◽  
Roger Whitney
Keyword(s):  
Generating Function ◽  
Lattice Paths ◽  
Plane Partitions ◽  
Bijective Proof ◽  
Reverse Plane

scholarly journals An Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula

1998 ◽  
Vol Vol. 3 no. 1 ◽  
Author(s):  
Christian Krattenthaler
Keyword(s):  
Generating Function ◽  
Plane Partitions ◽  
Bijective Proof ◽  
International Audience ◽  
Reverse Plane

International audience A bijective proof for Stanley's hook-content formula for the generating function for column-strict reverse plane partitions of a given shape is given that does not involve the involution principle of Garsia and Milne. It is based on the Hillman-Grassl algorithm and Schützenberger's \emphjeu de taquin.

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

2019 ◽  
Vol 168 ◽  
pp. 120-163
Author(s):  
Byung-Hak Hwang ◽  
Jang Soo Kim ◽  
Meesue Yoo ◽  
Sun-mi Yun
Keyword(s):  
Euler Numbers ◽  
Plane Partitions ◽  
Reverse Plane

A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2014 ◽  
Vol 91 (1) ◽  
pp. 41-46 ◽  
Author(s):  
ERNEST X. W. XIA
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Plane Partitions ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Congruence Modulo

AbstractFor any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$

scholarly journals Brane webs and random processes

2015 ◽  
Vol 30 (33) ◽  
pp. 1550202 ◽  
Author(s):  
Amer Iqbal ◽  
Babar A. Qureshi ◽  
Khurram Shabbir ◽  
Muhammad A. Shehper
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Generating Function ◽  
Random Processes ◽  
Open String ◽  
Plane Partitions ◽  
Schur Process ◽  
String Amplitudes

We study (p, q) 5-brane webs dual to certain N M5-brane configurations and show that the partition function of these brane webs gives rise to cylindric Schur process with period N. This generalizes the previously studied case of period 1. We also show that open string amplitudes corresponding to these brane webs are captured by the generating function of cylindric plane partitions with profile determined by the boundary conditions imposed on the open string amplitudes.

scholarly journals Enumeration of Cylindric Plane Partitions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Robin Langer
Keyword(s):  
Natural Generalization ◽  
Path Model ◽  
Plane Partition ◽  
Combinatorial Interpretation ◽  
Plane Partitions ◽  
Generating Series ◽  
International Audience ◽  
The Individual ◽  
The Right ◽  
Reverse Plane

International audience Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. As in the reverse plane partition case, the right hand side of this identity admits a simple factorization form in terms of the "hook lengths'' of the individual boxes in the underlying shape. The first result of this paper is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result of this paper is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog in terms of the non-intersecting lattice path model for cylindric plane partitions. Les partitions planes cylindriques sont une généralisation naturelle des partitions planes renversées. Une série génératrice pour énumération des partitions planes cylindriques a été donnée récemment par Borodin. Comme dans le cas des partitions planes renversées, la partie droite de cette identité peut être factoriser en terme de "longueur d’équerres'' des carrés dans la forme sous-jacente. Le premier résultat de cet article est une nouvelle preuve bijective de l'identité de Borodin qui utilise le cadre de "diagramme de croissance'' de Fomin pour la correspondance de RSK généralisée. Le deuxième résultat de cette article est une $(q,t)$-déformation d'identité de Borodin qui généralise un résultat de Okada dans le cas des partitions planes renversées. Le troisième résultat de cet article est une formule combinatoire explicite pour le poids de Macdonald qui utilise le modèle des chemins non-intersectant pour les partitions planes cylindriques.

scholarly journals Plane partitions and the combinatorics of some families of reduced Kronecker coefficients.

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Laura Colmenarejo
Keyword(s):  
Generating Function ◽  
Plane Partitions ◽  
Rate Of Growth ◽  
International Audience ◽  
Kronecker Coefficients

International audience We compute the generating function of some families of reduced Kronecker coefficients. We give a combi- natorial interpretation for these coefficients in terms of plane partitions. This unexpected relation allows us to check that the saturation hypothesis holds for the reduced Kronecker coefficients of our families. We also compute the quasipolynomial that govern these families, specifying the degree and period. Moving to the setting of Kronecker co- efficients, these results imply some observations related to the rate of growth experienced by the families of Kronecker coefficients associated to the reduced Kronecker coefficients already studied.

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'.

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