scholarly journals A Combinatorial Proof of the Recurrence for Rook Paths

10.37236/2105 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Emma Yu Jin ◽  
Markus E. Nebel
Keyword(s):  
Initial Conditions ◽  
Lattice Paths ◽  
Combinatorial Proof ◽  
Dyck Paths ◽  
Open Question

Let $a_n$ count the number of $2$-dimensional rook paths $\mathcal{R}_{n,n}$ from $(0,0)$ to $(2n,0)$. Rook paths $\mathcal{R}_{m,n}$ are the lattice paths from $(0,0)$ to $(m+n,m-n)$ with allowed steps $(x,x)$ and $(y,-y)$ where $x,y\in\mathbb{N}^{+}$. In answer to the open question proposed by M. Erickson et al. (2010), we shall present a combinatorial proof for the recurrence of $a_n$, i.e., $(n+1)a_{n+1}+9(n-1)a_{n-1}=2(5n+2)a_n$ with initial conditions $a_0=1$ and $a_1=2$. Furthermore, our proof can be extended to show the recurrence for the number of multiple Dyck paths $d_n$, i.e., $(n+2)d_{n+1}+9(n-1)d_{n-1}=5(2n+1)d_n$ with $d_0=1$ and $d_1=1$, where $d_n=\mathcal{N}_n(4)$ and $\mathcal{N}_n(x)$ is Narayana polynomial.

scholarly journals Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption

10.37236/1637 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
E. J. Janse van Rensburg ◽  
A. Rechnitzer
Keyword(s):  
Critical Point ◽  
Generating Functions ◽  
Asymptotic Expression ◽  
Path Model ◽  
Motzkin Path ◽  
Dyck Paths ◽  
Asymptotic Expressions ◽  
Directed Walks ◽  
Open Question ◽  
Vertex Colouring

In a previous work [26], by considering paths that are partially weighted, the generating function of Dyck paths was shown to possess a type of symmetry, called an exchange relation, derived from the exchange of a portion of the path between weighted and unweighted halves. This relation is particularly useful in solving for the generating functions of certain models of vertex-coloured Dyck paths; this is a directed model of copolymer adsorption, and in a particular case it is possible to find an asymptotic expression for the adsorption critical point of the model as a function of the colouring. In this paper we examine Motzkin path and partially directed walk models of the same adsorbing directed copolymer problem. These problems are an interesting generalisation of previous results since the colouring can be of either the edges, or the vertices, of the paths. We are able to find asymptotic expressions for the adsorption critical point in the Motzkin path model for both edge and vertex colourings, and for the partially directed walk only for edge colourings. The vertex colouring problem in partially directed walks seems to be beyond the scope of the methods of this paper, and remains an open question. In both these cases we first find exchange relations for the generating functions, and use those to find the asymptotic expression for the adsorption critical point.

scholarly journals Bijections for lattice paths between two boundaries

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sergi Elizalde ◽  
Martin Rubey
Keyword(s):  
Joint Distribution ◽  
Tutte Polynomial ◽  
Lattice Paths ◽  
Dyck Paths ◽  
International Audience ◽  
Pattern Avoiding Permutations

International audience We prove that on the set of lattice paths with steps $N=(0,1)$ and $E=(1,0)$ that lie between two boundaries $B$ and $T$, the two statistics `number of $E$ steps shared with $B$' and `number of $E$ steps shared with $T$' have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps $S=(0,-1)$ at prescribed $x$-coordinates. We also show that a similar equidistribution result for other path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. Finally, we extend the two theorems to $k$-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, and to pattern-avoiding permutations. On montre que, sur l'ensemble des chemins avec des pas $N=(0,1)$ et $E=(1,0)$ qui se trouvent entre deux chemins donnés $B$ et $T$, les deux statistiques"`nombre des pas $E$ en commun avec $B$" et "nombre des pas $E$ en commun avec $T$" ont une distribution conjointe symétrique. On donne une involution qui échange ces deux statistiques, préserve quelques autres paramètres additionnels, et admet une généralisation à des chemins avec des pas $S=(0, -1)$ dans des positions données. On montre aussi un autre résultat d'équidistribution similaire, lié au polynôme de Tutte d'un matroïde. Finalement, on étend les deux théorèmes à $k$-tuples de chemins entre deux frontières, et on donne quelques applications aux chemins de Dyck, en généralisant un résultat de Deutsch, et aux permutations avec des motifs exclus.

scholarly journals Enumerating Lattice Paths Touching or Crossing the Diagonal at a Given Number of Lattice Points

10.37236/2477 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michael Z. Spivey
Keyword(s):  
Lattice Points ◽  
Lattice Paths ◽  
Combinatorial Proof ◽  
Combinatorial Argument

We give bijective proofs that, when combined with one of the combinatorial proofs of the general ballot formula, constitute a combinatorial argument yielding the number of lattice paths from $(0,0)$ to $(n,rn)$ that touch or cross the diagonal $y = rx$ at exactly $k$ lattice points.  This enumeration partitions all lattice paths from $(0,0)$ to $(n,rn)$.  While the resulting formula can be derived using results from Niederhausen, the bijections and combinatorial proof are new.

scholarly journals On Parking Functions and the Zeta Map in Types B, C and D

10.37236/6714 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Robin Sulzgruber ◽  
Marko Thiel
Keyword(s):  
Hilbert Series ◽  
Root Systems ◽  
The Other ◽  
Lattice Paths ◽  
Square Lattice ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Coxeter Number ◽  
Dyck Paths ◽  
Diagonal Harmonics

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

scholarly journals Initial state perturbations in ensemble forecasting

2008 ◽  
Vol 15 (5) ◽  
pp. 751-759 ◽  
Author(s):  
L. Magnusson ◽  
E. Källén ◽  
J. Nycander
Keyword(s):  
Initial Conditions ◽  
Weather Prediction ◽  
Empirical Orthogonal Functions ◽  
Initial State ◽  
Orthogonal Functions ◽  
Ensemble Forecasts ◽  
Long Time ◽  
Chaotic Nature ◽  
Prediction Systems ◽  
Open Question

Abstract. Due to the chaotic nature of atmospheric dynamics, numerical weather prediction systems are sensitive to errors in the initial conditions. To estimate the forecast uncertainty, forecast centres produce ensemble forecasts based on perturbed initial conditions. How to optimally perturb the initial conditions remains an open question and different methods are in use. One is the singular vector (SV) method, adapted by ECMWF, and another is the breeding vector (BV) method (previously used by NCEP). In this study we compare the two methods with a modified version of breeding vectors in a low-order dynamical system (Lorenz-63). We calculate the Empirical Orthogonal Functions (EOF) of the subspace spanned by the breeding vectors to obtain an orthogonal set of initial perturbations for the model. We will also use Normal Mode perturbations. Evaluating the results, we focus on the fastest growth of a perturbation. The results show a large improvement for the BV-EOF perturbations compared to the non-orthogonalised BV. The BV-EOF technique also shows a larger perturbation growth than the SVs of this system, except for short time-scales. The highest growth rate is found for the second BV-EOF for the long-time scale. The differences between orthogonal and non-orthogonal breeding vectors are also investigated using the ECMWF IFS-model. These results confirm the results from the Loernz-63 model regarding the dependency on orthogonalisation.

scholarly journals Difference Equations and Generating Functions for some Lattice Path Problems

2019 ◽  
pp. 551-559
Author(s):  
Sreelatha Chandragiri
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Generating Functions ◽  
Lattice Paths ◽  
Lattice Path ◽  
Dyck Paths ◽  
Constant Coefficients ◽  
Novel Method

An identity for generating functions is proved in this paper. A novel method to compute the number of restricted lattice paths is developed on the basis of this identity. The method employs a difference equation with non-constant coefficients. Dyck paths, Schr¨oder paths, Motzkins path and other paths are computed to illustrate this method

scholarly journals Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules

10.37236/1821 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Nicholas A. Loehr ◽  
Jeffrey B. Remmel
Keyword(s):  
Generating Function ◽  
Hilbert Series ◽  
Lattice Paths ◽  
Weighted Sums ◽  
Parking Functions ◽  
Dyck Paths ◽  
Diagonal Harmonics ◽  
Combinatorial Formulas ◽  
New Statistics ◽  
Univariate Distribution

Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the Garsia-Haiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules.

scholarly journals Rational Associahedra and Noncrossing Partitions

10.37236/3432 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades ◽  
Nathan Williams
Keyword(s):  
Simplicial Complex ◽  
Rational Number ◽  
Perfect Matchings ◽  
Lattice Paths ◽  
Noncrossing Partitions ◽  
Positive Rational Number ◽  
Dyck Paths ◽  
Narayana Numbers ◽  
Positive Integers ◽  
Coprime Positive Integers

Each positive rational number $x>0$ can be written uniquely as $x=a/(b-a)$ for coprime positive integers $0<a<b$. We will identify $x$ with the pair $(a,b)$. In this paper we define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass}(x)=\mathsf{Ass}(a,b)$ called the rational associahedron.  It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the rational Catalan number $$\mathsf{Cat}(x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)!}{a!\,b!}.$$The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and Fomin-Reading.  We prove that $\mathsf{Ass}(a,b)$ is shellable and give nice product formulas for its $h$-vector (the rational Narayana numbers) and $f$-vector (the rational Kirkman numbers).  We define $\mathsf{Ass}(a,b)$ via rational Dyck paths: lattice paths from $(0,0)$ to $(b,a)$ staying above the line $y = \frac{a}{b}x$.  We also use rational Dyck paths to define a rational generalization of noncrossing perfect matchings of $[2n]$.  In the case $(a,b) = (n, mn+1)$, our construction produces the noncrossing partitions of $[(m+1)n]$ in which each block has size $m+1$.

scholarly journals Inversion Generating Functions for Signed Pattern Avoiding Permutations

10.37236/6545 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Naiomi T. Cameron ◽  
Kendra Killpatrick
Keyword(s):  
Generating Functions ◽  
Combinatorial Proof ◽  
Hyperoctahedral Group ◽  
Major Index ◽  
Signed Permutations ◽  
Mahonian Statistics ◽  
Almost All ◽  
Open Question ◽  
Pattern Avoiding Permutations

We consider the classical Mahonian statistics on the set $B_n(\Sigma)$ of signed permutations in the hyperoctahedral group $B_n$ which avoid all patterns in $\Sigma$, where $\Sigma$ is a set of patterns of length two.  In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\left|B_n(\Sigma)\right|$ is constant whenever $\left|\Sigma\right|=1$, whereas in most but not all instances where $\left|\Sigma\right|=2$, $\left|B_n(\Sigma)\right|=(n+1)!$.  We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\left|B_n(\Sigma)\right|=(n+1)!$.  In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on $B_n(21, \bar{2}\bar{1})$ and by giving the major index on $D_n(\Sigma)$ for $\Sigma =\{21, \bar{2}\bar{1}\}$ and $\Sigma=\{12,21\}$.  The main result of this paper is to give the inversion generating functions for $B_n(\Sigma)$ for almost all sets $\Sigma$ with $\left|\Sigma\right|\leq2.$

scholarly journals On the number of restricted Dyck paths

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 191-201
Author(s):  
Aleksandar Ilic ◽  
Andreja Ilic
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Integer Lattice ◽  
Lattice Paths ◽  
Mathematical Society ◽  
Spectral Moments ◽  
Estrada Index ◽  
Dyck Paths ◽  
Combinatorial Technique

In this note we examine the number of integer lattice paths consisting of up-steps (1, 1) and down-steps (1,?1) that do not touch the lines y = m and y=?k, and in particular Theorem 3.2 in [P. Mladenovic, Combinatorics, Mathematical Society of Serbia, Belgrade, 2001]. The theorem is shown to be incorrect for n ? m + k + min(m,k), and using similar combinatorial technique we proved the upper and lower bound for the number of such restricted Dyck paths. In conclusion, we present some relations between the Chebyshev polynomials of the second kind and generating function for the number of restricted Dyck paths, and connections with the spectral moments of graphs and the Estrada index.

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