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 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 Generating functions for the area below some lattice paths

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Lattice Paths ◽  
The Matrix ◽  
International Audience ◽  
Riordan Array

International audience We study some lattice paths related to the concept ofgenerating trees. When the matrix associated to this kind of trees is a Riordan array $D=(d(t),h(t))$, we are able to find the generating function for the total area below these paths expressed in terms of the functions $d(t)$ and $h(t)$.

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 Congruences modulo powers of 5 for the rank parity function

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Dandan Chen ◽  
Rong Chen ◽  
Frank Garvan
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Theta Function ◽  
Mock Theta Function ◽  
Parity Function ◽  
International Audience

International audience It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the rank parity function is f (q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

scholarly journals Counting words with Laguerre polynomials

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Equal Length ◽  
Combinatorial Interpretation ◽  
International Audience ◽  
Nous Utilisons

International audience We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of formal sums of Laguerre polynomials. We use this method to find the generating function for $k$-ary words avoiding any vincular pattern that has only ones. We also give generating functions for $k$-ary words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length, as well as the analogous results for compositions. Nous développons une méthode pour compter des mots satisfaisants certaines restrictions en établissant une interprétation combinatoire utile d’un produit de sommes formelles de polynômes de Laguerre. Nous utilisons cette méthode pour trouver la série génératrice pour les mots $k$-aires évitant les motifs vinculars consistant uniquement de uns. Nous présentons en suite les séries génératrices pour les mots $k$-aires évitant de façon cyclique les motifs vinculars consistant uniquement de uns et dont chaque série de uns entre deux tirets est de la même longueur. Nous présentons aussi les résultats analogues pour les compositions.

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.

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals Random sampling of lattice paths with constraints, via transportation

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Lucas Gerin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Sampling ◽  
Optimal Transport ◽  
Lattice Paths ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Contraction Property ◽  
International Audience

International audience We build and analyze in this paper Markov chains for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. These chains are easy to implement, and sample an "almost" uniform path of length $n$ in $n^{3+\epsilon}$ steps. This bound makes use of a certain $\textit{contraction property}$ of the Markov chain, and is proved with an approach inspired by optimal transport.

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