231 research outputs found
Old and young leaves on plane trees
A leaf of a plane tree is called an old leaf if it is the leftmost child of
its parent, and it is called a young leaf otherwise. In this paper we enumerate
plane trees with a given number of old leaves and young leaves. The formula is
obtained combinatorially by presenting two bijections between plane trees and
2-Motzkin paths which map young leaves to red horizontal steps, and old leaves
to up steps plus one. We derive some implications to the enumeration of
restricted permutations with respect to certain statistics such as pairs of
consecutive deficiencies, double descents, and ascending runs. Finally, our
main bijection is applied to obtain refinements of two identities of Coker,
involving refined Narayana numbers and the Catalan numbers.Comment: 11 pages, 7 figure
Path counting and random matrix theory
We establish three identities involving Dyck paths and alternating Motzkin
paths, whose proofs are based on variants of the same bijection. We interpret
these identities in terms of closed random walks on the halfline. We explain
how these identities arise from combinatorial interpretations of certain
properties of the -Hermite and -Laguerre ensembles of random
matrix theory. We conclude by presenting two other identities obtained in the
same way, for which finding combinatorial proofs is an open problem.Comment: 14 pages, 13 figures and diagrams; submitted to the Electronic
Journal of Combinatoric
Crossings, Motzkin paths and Moments
Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain
-analogues of Laguerre and Charlier polynomials. The moments of these
orthogonal polynomials have combinatorial models in terms of crossings in
permutations and set partitions. The aim of this article is to prove simple
formulas for the moments of the -Laguerre and the -Charlier polynomials,
in the style of the Touchard-Riordan formula (which gives the moments of some
-Hermite polynomials, and also the distribution of crossings in matchings).
Our method mainly consists in the enumeration of weighted Motzkin paths, which
are naturally associated with the moments. Some steps are bijective, in
particular we describe a decomposition of paths which generalises a previous
construction of Penaud for the case of the Touchard-Riordan formula. There are
also some non-bijective steps using basic hypergeometric series, and continued
fractions or, alternatively, functional equations.Comment: 21 page
Noncommutative integrability, paths and quasi-determinants
In previous work, we showed that the solution of certain systems of discrete
integrable equations, notably and -systems, is given in terms of
partition functions of positively weighted paths, thereby proving the positive
Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of
solution is amenable to generalization to non-commutative weighted paths. Under
certain circumstances, these describe solutions of discrete evolution equations
in non-commutative variables: Examples are the corresponding quantum cluster
algebras [BZ], the Kontsevich evolution [DFK09b] and the -systems themselves
[DFK09a]. In this paper, we formulate certain non-commutative integrable
evolutions by considering paths with non-commutative weights, together with an
evolution of the weights that reduces to cluster algebra mutations in the
commutative limit. The general weights are expressed as Laurent monomials of
quasi-determinants of path partition functions, allowing for a non-commutative
version of the positive Laurent phenomenon. We apply this construction to the
known systems, and obtain Laurent positivity results for their solutions in
terms of initial data.Comment: 46 pages, minor typos correcte
Chebyshev type lattice path weight polynomials by a constant term method
We prove a constant term theorem which is useful for finding weight
polynomials for Ballot/Motzkin paths in a strip with a fixed number of
arbitrary `decorated' weights as well as an arbitrary `background' weight. Our
CT theorem, like Viennot's lattice path theorem from which it is derived
primarily by a change of variable lemma, is expressed in terms of orthogonal
polynomials which in our applications of interest often turn out to be
non-classical. Hence we also present an efficient method for finding explicit
closed form polynomial expressions for these non-classical orthogonal
polynomials. Our method for finding the closed form polynomial expressions
relies on simple combinatorial manipulations of Viennot's diagrammatic
representation for orthogonal polynomials. In the course of the paper we also
provide a new proof of Viennot's original orthogonal polynomial lattice path
theorem. The new proof is of interest because it uses diagonalization of the
transfer matrix, but gets around difficulties that have arisen in past attempts
to use this approach. In particular we show how to sum over a set of implicitly
defined zeros of a given orthogonal polynomial, either by using properties of
residues or by using partial fractions. We conclude by applying the method to
two lattice path problems important in the study of polymer physics as models
of steric stabilization and sensitized flocculation.Comment: 27 pages, 14 figure
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