scholarly journals Combinatorics of the Three-Parameter PASEP Partition Function

10.37236/509 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Matthieu Josuat-Vergès
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Exclusion Process ◽  
Asymmetric Exclusion Process ◽  
Combinatorial Interpretation ◽  
Rook Placements ◽  
Combinatorial Interpretations ◽  
The Matrix ◽  
The One ◽  
New Formula

We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a generating function of permutation tableaux by the combinatorial interpretation of Corteel and Williams. We prove bijectively two new combinatorial interpretations. The first one is in terms of weighted Motzkin paths called Laguerre histories and is obtained by refining a bijection of Foata and Zeilberger. Secondly we show that this partition function is the generating function of permutations with respect to right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by refining a bijection of Françon and Viennot. Then we give a new formula for the partition function which generalizes the one of Blythe & al. It is proved in two combinatorial ways. The first proof is an enumeration of lattice paths which are known to be a solution of the Matrix Ansatz of Derrida & al. The second proof relies on a previous enumeration of rook placements, which appear in the combinatorial interpretation of a related normal ordering problem. We also obtain a closed formula for the moments of Al-Salam-Chihara polynomials.

scholarly journals Matrix Ansatz, lattice paths and rook placements

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
S. Corteel ◽  
M. Josuat-Vergès ◽  
T. Prellberg ◽  
M. Rubey
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Laguerre Polynomials ◽  
Lattice Paths ◽  
Rook Placements ◽  
Combinatorial Interpretations ◽  
The Matrix ◽  
International Audience ◽  
Enumeration Formula

International audience We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrences of the pattern $13-2$, the generating function according to weak exceedances and crossings, and the $n^{\mathrm{th}}$ moment of certain $q$-Laguerre polynomials. Nous donnons deux interprétations combinatoires du Matrix Ansatz du PASEP en termes de chemins et de placements de tours. Cela donne deux preuves (presque) combinatoires d'une nouvelle formule pour la fonction de partition du PASEP. Cette formule donne aussi par exemple la fonction génératrice des permutations de taille donnée par rapport au nombre de montées et d'occurrences du motif $13-2$, la fonction génératrice par rapport au nombre d'excédences faibles et de croisements, et le $n^{\mathrm{ième}}$ moment de certains polynômes de $q$-Laguerre.

scholarly journals Multivariate Eulerian Polynomials and Exclusion Processes

2016 ◽  
Vol 25 (4) ◽  
pp. 486-499 ◽  
Author(s):  
P. BRÄNDÉN ◽  
M. LEANDER ◽  
M. VISONTAI
Keyword(s):  
Finite Number ◽  
Exclusion Process ◽  
Partition Functions ◽  
Asymmetric Exclusion Process ◽  
Negative Dependence ◽  
Combinatorial Interpretation ◽  
Exclusion Processes ◽  
Eulerian Polynomials ◽  
The Third ◽  
Dependence Properties

We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and coloured permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalisations of Eulerian polynomials for coloured permutations considered recently by N. Williams and the third author, and others. We also discuss stability and negative dependence properties satisfied by the partition functions.

Arithmetic properties of l-regular overpartitions

2016 ◽  
Vol 12 (03) ◽  
pp. 841-852 ◽  
Author(s):  
Erin Y. Y. Shen
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Combinatorial Interpretation ◽  
Arithmetic Properties ◽  
Vector Partitions

Recently, Andrews introduced the partition function [Formula: see text] as the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. He proved that [Formula: see text] and [Formula: see text] are divisible by [Formula: see text]. Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we call the overpartitions enumerated by the function [Formula: see text] [Formula: see text]-regular overpartitions. For [Formula: see text] and [Formula: see text], we obtain some explicit results on the generating function dissections. We also derive some congruences for [Formula: see text] modulo [Formula: see text], [Formula: see text] and [Formula: see text] which imply the congruences for [Formula: see text] proved by Andrews. By introducing a rank of vector partitions, we give a combinatorial interpretation of the congruences of Andrews for [Formula: see text] and [Formula: see text].

scholarly journals The topological line of ABJ(M) theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Nicola Gorini ◽  
Luca Griguolo ◽  
Luigi Guerrini ◽  
Silvia Penati ◽  
Domenico Seminara ◽  
...  
Keyword(s):  
Partition Function ◽  
Central Charge ◽  
Exact Representation ◽  
One Dimensional ◽  
Topological Sector ◽  
The Matrix ◽  
The One ◽  
The Masses ◽  
Dimension One ◽  
M Theory

Abstract We construct the one-dimensional topological sector of $$ \mathcal{N} $$ N = 6 ABJ(M) theory and study its relation with the mass-deformed partition function on S3. Supersymmetric localization provides an exact representation of this partition function as a matrix integral, which interpolates between weak and strong coupling regimes. It has been proposed that correlation functions of dimension-one topological operators should be computed through suitable derivatives with respect to the masses, but a precise proof is still lacking. We present non-trivial evidence for this relation by computing the two-point function at two-loop, successfully matching the matrix model expansion at weak coupling and finite ranks. As a by-product we obtain the two-loop explicit expression for the central charge cT of ABJ(M) theory. Three- and four-point functions up to one-loop confirm the relation as well. Our result points towards the possibility to localize the one-dimensional topological sector of ABJ(M) and may also be useful in the bootstrap program for 3d SCFTs.

THE RESTRICTED 3-COLORED PARTITION FUNCTION MOD 3

2013 ◽  
Vol 09 (07) ◽  
pp. 1789-1799
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Modular Forms ◽  
Type Identity ◽  
Colored Partitions ◽  
Combinatorial Interpretations

In this paper, we investigate the divisibility of the function b(n), counting the number of certain restricted 3-colored partitions of n. We obtain one Ramanujan type identity, which implies that b(3n + 2) ≡ 0 ( mod 3). Furthermore, we study the generating function for b(3n + 1) by modular forms. Finally, we find two cranks as combinatorial interpretations of the fact that b(3n + 2) is divisible by 3 for any n.

VIRASORO CONSTRAINTS ON THE MATRIX-MODEL PARTITION FUNCTION AND STRING FIELD THEORY

1992 ◽  
Vol 07 (07) ◽  
pp. 1553-1581 ◽  
Author(s):  
ASHOKE SEN
Keyword(s):  
Field Theory ◽  
Partition Function ◽  
Matrix Model ◽  
String Field Theory ◽  
Minimal Model ◽  
Differential Operators ◽  
Tree Level ◽  
Virasoro Constraints ◽  
The Matrix ◽  
The One

The one-matrix model at the kth multicritical point is known to describe the (2, 2k–1) minimal model coupled to gravity, and the partition function of this model is known to obey a set of Virasoro constraints generated by a set of differential operators Ln. Working at the tree level of string theory, and using the Feigin-Fuchs description of the (2, 2k–1) minimal model, we show that the Virasoro constraints generated by Li (0≤i≤k−2) can be identified with a set of gauge symmetries of the corresponding string field theory.

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