The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size -3,-5,-7,... . For such paths, we find the generating functions of them, according to length, ending at level $i$, both, when considering them from left to right and from right to left. The generating functions are intrinsically cubic, and thus (for $i=0$) in bijection to various objects, like even trees, ternary trees, etc.
A variation of Dyck paths allows for down-steps of arbitrary length, not just one. This is motivated by ideas due to Emeric Deutsch. We use the adding-a-new-slice technique and the kernel method to compute the number of maximal runs of up-step runs of length 1 and a subclass of Deutsch paths satisfying a condition that was stipulated by R. Stanley for Dyck paths.
A variation of Dyck paths allows for down-steps of arbitrary length, not just one. Credits for this invention are given to Emeric Deutsch. Surprisingly, the enumeration of them is somewhat akin to the analysis of Motzkin-paths; the last section contains a bijection.
A variation of Dyck paths allows for down-steps of arbitrary length, not just one. This is motivated by ideas published by Emeric Deutsch around the turn of the millenium. We are interested in the subclass of them where the sequence of the levels of valleys is non-decreasing. This was studied around 20 years ago in the classical case.
This paper is motivated by two problems recently proposed by Coker on combinatorial identities related to the Narayana polynomials and the Catalan numbers. We find that a bijection of Chen, Deutsch and Elizalde can be used to provide combinatorial interpretations of the identities of Coker when it is applied to weighted plane trees. For the sake of presentation of our combinatorial correspondences, we provide a description of the bijection of Chen, Deutsch and Elizalde in a slightly different manner in the form of a direct construction from plane trees to 2-Motzkin paths without the intermedia…
Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra was an essential tool. Some results are new, some have appeared before. The lattice path models we treated, are: Hoppy's walks, the combinatorics of sequence A002212 in \cite{OEIS} (skew Dyck paths, Schröder paths, Hex-trees, decorated ordered trees, multi-edge trees, etc.) Weighted unary-binary trees also occur, and we could improve on our old paper on Horton-Strahler numbers \cite{FlPr86}, by using…
We consider a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ down-steps $(1,-j)$, for $j\ge2$ are allowed. We give credits to Emeric Deutsch for that. The enumeration of such objects living in a strip is performed. Methods are the kernel method and techniques from linear algebra.
School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, Institut Teknologi Bandung（万隆理工学院数学系）；Department of IT and Big Data Analytics, Calvin Institute of Technology（卡尔文理工学院信息技术与大数据分析系）
In 2003, Deutsch and Elizalde defined a family of bijective maps between the set of Dyck paths to itself which is induced by some particular permutations. In this paper, we extend the construction of the maps by allowing the permutation to be arbitrary. We characterise the permutations which generate the same map and find all permutations generating a bijection among Dyck paths. Consequently, we give a new combinatorial interpretation of the quantity $(2n-1)!!$ as well as some new statistics of Dyck paths which are equidistributed to some known height statistics via our generalised maps.
We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics `number of E steps shared with B' and `number of E steps shared with T' have a symmetric joint distribution. To do so, 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 path statistics follows from the fact that the Tutte polynomial of a matroid is inde…
A redundant generating function is a generating function having terms which are not part of the solution of the original problem. We use redundant generating functions to study two path problems. In the first application we explain a surprising occurrence of Catalan numbers in counting paths that stay below the line y = 2x. In the second application we prove a conjecture of Niederhausen and Sullivan.
A plateau in a Motzkin path is a sequence of three steps: an up step, a horizontal step, then a down step. We find three different forms for the bivariate generating function for plateaus in Motzkin paths, then generalize to longer plateaus. We conclude by describing a further generalization: a continued fraction form from which one can easily derive new multivariate generating functions for various kinds of path statistics. Several examples of generating functions are given using this technique.
The question of classifying the nature of the generating functions of restricted lattice walks has enjoyed much attention in past years. We prove that a certain class of octant walks have a D-finite generating function using the theory of multivariate formal Laurent series.
Using a recursive approach, we show that the generating function for sets of Motzkin paths avoiding a single (not necessarily consecutive) pattern is rational over $x$ and the Catalan generating function $C(x) = \frac{1-\sqrt{1-4x^2}}{2x^2}$, where $x$ keeps track of the length of the path. Moreover, an algorithm is provided for finding the generating function in the more general case of an arbitrary set of patterns. In addition, this algorithm allows us to find a combinatorial specification for pattern-avoiding Motzkin paths, which can be used not only for enumeration, but also for exhaustive…
We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant evaluations have numerous corollaries. In particular, one consequence is that the Hankel determinant of Motzkin prefix numbers equals 1, regardless of the size of the Hankel matrix.
We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers - most notably of Catalan, ballot, and of Motzkin numbers - that appeared previously in the literature.
In a recent preprint, Lai and Rohatgi compute the generating functions of lozenge tilings of "quartered hexagons with dents" by applying the method of "graphical condensation". The purpose of this note is to exhibit how (a generalization of) Theorems 2.1 and 2.2 in Lai and Rohatgi's preprint can be achieved by the Lindström--Gessel--Viennot method of non--intersecting lattice paths and a certain determinant evaluation.
In a recent preprint, Lai worked out the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents" which differ only in width. Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindström--Gessel--Viennot method for nonintersecting lattice paths in a quite simple way. Basically the same observation, but restricted to mere enumeration (i.e., all weights of lozenge tilings are equal to $1$), i…
In a recent preprint, Lai showed that the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents", which differ only in width, factors nicely, and the same is true for the quotient of generating functions of weighted lozenge tilings of two "quarter hexagons with lateral dents". Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindström--Gessel--Viennot method for nonintersecting lattice …
We consider paths in the plane with $(1,0),$ $(0,1),$ and $(a,b)$-steps that start at the origin, end at height $n,$ and stay to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most $b/a,$ then the ordinary generating function for the number of such paths ending at height $n$ is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or ``umbral'' generating function, in which the power $z^n$ is replaced by a power series of the form $z^n ϕ_n(z),$ where $ϕ_n(0) = 1.$ Then we c…
A Schröder path is a lattice path from $(0,0)$ to $(2n,0)$ with steps $(1,1)$, $(1,-1)$ and $(2,0)$ that never goes below the $x-$axis. A small Schröder path is a Schröder path with no $(2,0)$ steps on the $x-$axis. In this paper, a 3-variable generating function $R_L(x,y,z)$ is given for Schröder paths and small Schröder paths respectively. As corollaries, we obtain the generating functions for several kinds of generalized Schröder paths counted according to the order in a unified way.
Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schroeder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight 2. This work presents bijections from those Dyck paths to little Schroeder paths, and from a similar set of Dyck paths to big Schroeder paths. We show the effect of these bijections on the corresponding matchings, find generating functions for two new classes of lattice paths, and demonstrate a relationship with 231-avoiding permutations.
A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths. The aim of this article is to extend the notion of local types from permutations to $k$-Stirling permutations (also known as $k$-multipermutati…
Given two relatively prime positive integers $α$ and $β$, we consider simple lattice paths (with unit East and unit North steps) from $(0,0)$ to $(αk,βk)$, and enumerate them by their left and right bounces with respect to the line $y=\fracβα x$. We give the corresponding multivariate generating functions for all such paths as well as for subclasses of paths that start and end with a prescribed step. For illustration purposes, we discuss the case $β=1$ and express some of our functions in terms of the Fuss-Catalan generating function $c_α(x)$.
In this note, we introduce a statistic on Motzkin paths that describes the rank generating function of Bruhat order for involutions. Our proof relies on a bijection introduced by Philippe Biane from permutations to certain labeled Motzkin paths and a recently introduced interpretation of this rank generating function in terms of visible inversions. By restricting our identity to fixed-point-free (FPF) involutions, we recover an identity due to Louis Billera, Lionel Levine and Karola Mészáros with a previous bijective proof by Matthew Watson. Our work sheds new light on the Ethiopian dinner gam…
A recurrence relation of the generating function of the dimer model of Fibonacci type gives a functional relation for formal power series associated to lattice paths such as a Dyck, Motzkin and Schröder path. In this paper, we generalize the correspondence to the case of generalized lattice paths, $k$-Dyck, $k$-Motzkin and $k$-Schröder paths, by modifying the recurrence relation of the dimer model. We introduce five types of generalizations of the dimer model by keeping its combinatorial structures. This allows us to express the generating functions in terms of generalized lattice paths. The w…
We provide formulas for generating functions of many types of paths in various rooted tree structures. We compute the $k$th moment of the generating functions for various types of vertical paths. In two specific familes of trees we find exact closed formulas for expectations and their asymptotic values. Some of these closed formulas are surprisingly simple.
In this thesis we develop generalized versions of the Chung-Feller theorem for lattice paths constrained in the half plane. The beautiful cycle method which was developed by Devoretzky and Motzkin as a means to prove the ballot problem is modified and applied to generalize the classical Chung-Feller theorem. We use Lagrange inversion to derive the generalized formulas. For the generating function proof we study various ways of decomposing lattice paths. We also show some results related to equidistribution properties in terms of Narayana and Catalan generating functions. We then develop genera…
A rook path is a path on lattice points in the plane in which any proper horizontal step to the right or vertical step north is allowed. If, in addition, one allow bishop steps, that is, proper diagonal steps of slope 1, then one has queen paths. A rook or queen path is Catalan if it starts at the origin and stays strictly to the left of the line y = x-1. We give explicit formulas for the ordinary generating function of the number of Catalan rook and queen paths finishing at $(n,n).$ These generating functions are algebraic; indeed, they satisfy quadratic equations. In the second version, we a…
A Dyck path is a lattice path in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of steps (1,1) and (1,-1), which never passes below the x-axis. A peak at height k on a Dyck path is a point on the path with coordinate y=k that is immediately preceded by a (1,1) step and immediately followed by a (1,-1) step. In this paper we find an explicit expression to the generating function for the number of Dyck paths starting at (0,0) and ending at (2n,0) with exactly r peaks at height k. This allows us to express this function via Chebyshev polynomials of the second kind and generatin…
This note collects some results and conjectures for the generating functions of the Hankel determinants of certain polynomials which are related to Motzkin paths.
We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for $b=o\left(\sqrt{n}\right)$, Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a $(1:b)$ game on $E(K_n)$. As another application, we show that for $b=Θ\left(n/\ln n\right)$, playing a $(1:b)$ game on $E(K_n)$, Maker can build a graph which contains copies of all sp…
We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.
Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks
Dartmouth College（达特茅斯学院）；Concordia University（康考迪亚大学）；The George Washington University（乔治华盛顿大学）；Willamette University（威拉米特大学）；North Dakota State University（北达科他州立大学）；Eastern Illinois University（东伊利诺伊大学）
In this work, we establish a general relationship between the enumeration of weighted directed paths and skew Schur functions, extending work by Bousquet-Mélou, who expressed generating functions of discrete excursions in terms of rectangular Schur functions.
Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic equations for both paths, and thus generating functions for the paths, are provided. Using these, various parameters involving the two paths are analyzed.
In a Dyck path a peak which is (weakly) higher than all the preceding peaks is called a strict (weak) left to right maximum. We obtain explicit generating functions for both weak and strict left to right maxima in Dyck paths. The proofs of the associated asymptotics make use of analytic techniques such as Mellin transforms, singularity analysis and formal residue calculus.
We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed graphs. As a particular case, we obtain a relation linking the Hamiltonian paths and cycles of a graph to its dominating connected sets.
Lattice paths in the quarter plane have led to a large and varied set of results in recent years. One major project has been the classification of step sets according to the properties of the corresponding generating functions, and this has involved a variety of techniques, some highly intricate and specialised. The famous Kreweras and reverse Kreweras walk models are two particularly interesting models, as they are among the only four cases which have algebraic generating functions. Here we investigate how the properties of the Kreweras and reverse Kreweras models change when boundary interac…
Consider a single walker on the slit plane, that is, the square grid Z^2 without its negative x-axis, who starts at the origin and takes his steps from a given set S. Mireille Bousquet-Melou conjectured that -- excluding pathological cases -- the generating function counting the number of possible walks is algebraic if and only if the walker cannot cross the negative x-axis without touching it. In this paper we prove a special case of her conjecture.
An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableau of various shapes $λ$ and contents $μ$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableau that "standardi…
Generating 2-Gray codes for grand Motzkin paths and grand Dyck paths with air pockets in constant amortized time
School of Computer Science and Engineering, Sun Yat-sen University（中山大学计算机科学与工程学院）；Faculty of Applied Sciences, Macao Polytechnic University（澳门理工大学应用科学学院）
We present an extension of a theorem by Michael Drmota and Michèle Soria [Images and Preimages in Random Mappings, 1997] that can be used to identify the limiting distribution for a class of combinatorial schemata. This is achieved by determining analytical and algebraic properties of the associated bivariate generating function. We give sufficient conditions implying a half-normal limiting distribution, extending the known conditions leading to either a Rayleigh, a Gaussian, or a convolution of the last two distributions. We conclude with three natural appearances of such a limiting distribut…
We define and study multi-colored dimer models on a segment and on a circle. The multivariate generating functions for the dimer models satisfy the recurrence relations similar to the one for Fibonacci numbers. We give closed formulae for the generating functions. We show that, in the large size limit with specializations of the formal variables, the generating functions exhibit the summations appearing in generalized Rogers--Ramanujan identities. Further, the generating functions of the dimer models have infinite product formulae for general values of formal variables in the large size limit.…
Using the generalized method of steepest descents for the case of two coalescing saddle points, we derive an asymptotic expression for the bivariate generating function of Dyck paths, weighted according to their length and their area in the limit of the area generating variable tending towards 1. The result is valid uniformly for a range of the length generating variable, including the tricritical point of the model.
Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \emph{discovery and proof} of the fact that the generating series $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: \[ G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.\]
In this paper we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory -- we find explicit formulae for classical generating functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We then focus on quivers corresponding to $(r,s)$ torus knots and show that their classical generating functions, in the extremal limit and framing $rs$, are generating functions of lattice paths under the line of the slope $r/s$. Generating functions of such paths satisfy extremal A-polynom…
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-th moment of certain q-Laguerre polynomials.
We derive the length and area generating function of planar height-restricted forward-moving discrete paths of increments +1, 0, or -1 with arbitrary starting and ending points, the so-called Motzkin meanders, and the more general length-area generating functions for Motzkin paths with markers monitoring the number of passages from the two height boundaries ('floor' and 'ceiling') and the time spent there. The results are obtained by embedding Motzkin paths in a two-step anisotropic Dyck path process and using propagator, exclusion statistics and bosonization techniques. We also present a clus…
We consider a generalised version of Motzkin paths, where horizontal steps have length $\ell$, with $\ell$ being a fixed positive integer. We first give the general functional equation for the area-length generating function of this model. Using a heuristic ansatz, we derive the area-length scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schröder paths, before generalising our approach to arbitrary $\ell$. We then derive an expression for the generating function of Schröder paths and analyse the scaling behaviour of this fu…
We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an …
