We count a large class of lattice paths by using factorizations of free monoids. Besides the classical lattice paths counting problems related to Catalan numbers, we give a new approach to the problem of counting walks on the slit plane (walks avoid a half line) that was first solved by Bousquet-Mélou and Schaeffer. We also solve a problem about walks in the half plane avoiding a half line by subsequently applying the factorizations of two different Gessel pairs, giving a generalization of a result of Bousquet-Mélou.
Gessel's walks are the planar walks that move within the positive quadrant $\mathbb{Z}_{+}^{2}$ by unit steps in any of the following directions: West, North-East, East and South-West. In this paper, we find an explicit expression for the trivariate generating function counting the Gessel's walks with $k\geq 0$ steps, which start at $(0,0)$ and end at a given point $(i,j) \in \mathbb{Z}^2_+$.
Gessel walks are lattice paths confined to the quarter plane that start at the origin and consist of unit steps going either West, East, South-West or North-East. In 2001, Ira Gessel conjectured a nice closed-form expression for the number of Gessel walks ending at the origin. In 2008, Kauers, Koutschan and Zeilberger gave a computer-aided proof of this conjecture. The same year, Bostan and Kauers showed, again using computer algebra tools, that the complete generating function of Gessel walks is algebraic. In this article we propose the first "human proofs" of these results. They are derived …
We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number of ways of walking $2n$ steps in the region $x+y \geq 0, y \geq 0$ of the square-lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals $16^n\frac{(5/6)_n(1/2)_n}{(5/3)_n(2)_n}$ .
In his Ph.D. thesis, Ira Gessel proved a reciprocity formula for noncommutative symmetric functions which enables one to count words and permutations with restrictions on the lengths of their increasing runs. We generalize Gessel's theorem to allow for a much wider variety of restrictions on increasing run lengths, and use it to complete the enumeration of permutations with parity restrictions on peaks and valleys, and to give a systematic method for obtaining generating functions for permutation statistics that are expressible in terms of increasing runs. Our methods can also be used to obtai…
We deduce Narayana's formula for the number of lattice paths that fit in a Young diagram as a direct consequence of the Gessel-Viennot theorem on non-intersecting lattice paths.
We consider four examples of short step lattice paths confined to the quarter plane. These are the Kreweras, Reverse Kreweras, Gessel, and Mishna-Rechnitzer lattice paths.The Reverse Kreweras are straightforward to solve and thus interesting as a contrast to the Kreweras paths and Gessel paths as the latter two have historically been significantly more difficult to solve. The Mishna-Rechnitzer paths are interesting as they are associated with an infinite order group. We will give some geometrical insight into all these properties by considering the Weyl chambers associated with their step sets…
The Gessel number $P(n,r)$ is the number of the paths in plane with $(1, 0)$ and $(0,1)$ steps from $(0,0)$ to $(n+r, n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x\geq r\}$. We show that there is a close relationship between the Gessel numbers $P(n,r)$ and the super Catalan numbers $S(n,r)$. By using new sums, we prove that an alternating convolution of the Gessel numbers $P(n,r)$ is always divisible by \frac{1}{2}$S(n,r)$.
The Gessel number $P(n,r)$ represents the number of lattice paths in a plane with unit horizontal and vertical steps from $(0,0)$ to $(n+r,n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x \geq r\}$. In this paper, we use combinatorial arguments to derive a recurrence relation between $P(n,r)$ and $P(n-1,r+1)$. Also, we give a new proof for a well-known closed formula for $P(n,r)$. Moreover, a new combinatorial interpretation for the Gessel numbers is presented.
We state and prove several theorems that demonstrate how the coordinate Bethe Ansatz for the eigenvectors of suitable transfer matrices of a generalised inhomogeneous five-vertex model on the square lattice, given certain conditions hold, is equivalent to the Gessel-Viennot determinant for the number of configurations of $N$ non-intersecting directed lattice paths, or vicious walkers, with various boundary conditions. Our theorems are sufficiently general to allow generalisation to any regular planar lattice.
A permutation whose any prefix has no more descents than ascents is called a ballot permutation. In this paper, we present a decomposition of ballot permutations that enables us to construct a bijection between ballot permutations and odd order permutations, which proves a set-valued extension of a conjecture due to Spiro using the statistic of peak values. This bijection also preserves the neighbors of the largest letter in permutations and thus resolves a refinement of Spiro' s conjecture proposed by Wang and Zhang. Our decomposition can be extended to well-labelled positive paths, a class o…
We interpret walks in the first quadrant with steps {(1,1),(1,0),(-1,0), (-1,-1)} as a generalization of Dyck words with two sets of letters. Using this language, we give a formal expression for the number of walks in the steps above beginning and ending at the origin. We give an explicit formula for a restricted class of such words using a correspondance between such words and Dyck paths. This explicit formula is exactly the same as that for the degree of the polynomial satisfied by the square of the area of cyclic n-gons conjectured by Dave Robbins although the connection is a mystery. Final…
We show how a formula concerning ``vicious walkers'' (which basically are nonintersecting lattice paths) on the cylinder given by P.J. Forrester can be proved and generalized by using the Lindström--Gessel--Viennot method, after having things set up in the right way. We apply the corresponding results to the (thermodynamic limit of the) free energy of the ``lock step model of vicious walkers'', thus completing (and in one instance correcting) the work of Forrester . Moreover, we also show how a related formula given by I. Gessel and C. Krattenthaler can be obtained from the same ``point of vie…
Suppose 2n voters vote sequentially for one of two candidates. For how many such sequences does one candidate have strictly more votes than the other at each stage of the voting? The answer is \binom{2n}{n} and, while easy enough to prove using generating functions, for example, only two combinatorial proofs exist, due to Kleitman and Gessel. In this paper we present two new (far simpler) bijective proofs.
A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.
For a single value of $\ell$, let $f(n,\ell)$ denote the number of lattice paths that use the steps $(1,1)$, $(1,-1)$, and $(\ell,0)$, that run from $(0,0)$ to $(n,0)$, and that never run below the horizontal axis. Equivalently, $f(n,\ell)$ satisfies the quadratic functional equation $F(x) = \sum_{n\ge 0}f(n,\ell) x^n = 1+x^{\ell}F(x)+x^2F(x)^2.$ Let $H_n$ denote the $n$ by $n$ Hankel matrix, defined so that $[H_n]_{i,j} = f(i+j-2,\ell)$. Here we investigate the values of such determinants where $\ell = 0,1,2,3$. For $\ell = 0,1,2$ we are able to employ the Gessel-Viennot-Lindström method. For…
In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström-Gessel-Viennot interpretation of semistandard Young tableaux and the Jacobi-Trudi identity together with elementary observations. After some preparations, this point of view provides very simple "graphical proofs" for classical determinantal identities like the Cauchy--Binet formula, Dodgson's condensation formula, the Plücker relations and Laplace's expansion. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which m…
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.
We give a combinatorial interpretation of a certain positivity conjecture of Chern-Schwartz-MacPherson classes, as stated by P. Aluffi and the author in a previous paper. It translates into a positivity property for a sum of p by p determinants consisting of binomial coefficients, generalizing the classical Theorem of Lindstrom-Gessel-Viennot et al. which computes these determinants in terms of non-intersecting lattice paths. We prove this conjecture for p=2,3.
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…
We show connection between Dyck paths with peaks of bounded height and random walks. The correspondence between a certain class of random walks and such Dyck paths allows us to develop a probabilistic perspective on Chebyshev polynomials.
We construct a combinatorial function F which computes the number of oriented Hamiltonian paths of any given type, in a transitive tournament. We also study many properties of F that arise, and reach some observations.
Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two examples, Webb (2004) introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy-Binet formula. This paper studies counting on linear matroid problems extending Webb's work. We first introduce "Pfaffian parities" as an extension of Pfaffian pairs to the linear matroid parity problem, which is a commo…
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 …
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.
In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schröder paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schröder paths, we also get an identity involving produc…
We revisit $R$-polynomials with introducing the new idea ``shifted $R$-polynomials" (or Bruhat weight) for all Bruhat intervals in finite Coxeter groups. Then, we apply these polynomials to weighted counting of Bruhat paths. Further, we prove a new criterion of irregularity of lower intervals as analogy of Carrell-Peterson's and Dyer's results. Also, we present the upper bound of shifted $R$-polynomials for Bruhat intervals of fixed length by Jacobsthal numbers.
A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path $D$, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly $D$. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.
Dyck paths are one of the most important objects in enumerative combinatorics, and there are many papers devoted to counting selected families of Dyck paths. Here we present two approaches for the automatic counting of many such families, using both a "dumb" approach (driven by numeric dynamic programming) that often works in practice, and a "clever" approach, needed for larger problems, driven by "symbolic" dynamic programming. Both approaches are fully automated and implemented in Maple.
Motzkin paths are simple yet important combinatorial objects. In this paper, we consider families of Motzkin paths with restrictions on peak heights, valley heights, upward-run lengths, downward-run lengths, and flat-run lengths. This paper presents two fully automated methods for enumerating the paths of such families. The first method uses numeric dynamic programming. While this method often times works, it can be slow and may not work for larger problems. The second method uses symbolic dynamic programming to solve such problems. These methods are implemented in the maple packages accompany…
Recently, Mansour and Shattuck related the total number of humps in all of the $(k, a)$-paths of order $n$ to the number of super $(k, a)$-paths, which generalized previous results concerning the cases when $k = 1$ and $a = 1$ or $a = \infty$. They also derived a relation on the total number of peaks in all of the $(k, a)$-paths of order $n$ and the number of super $(k, a)$-paths, and asked for bijective proofs. In this paper, we will give bijective proofs of these two relations.
The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from $(0,0)$ to $(n,n)$ which is below the diagonal line $y=x$. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from $(0,0)$ to $(m,n) \in \mathbb{N}^2$ which is below the diagonal line $y=\frac{n}{m}x$, and denote by $C(m,n)$ the number of Dyck paths from $(0,0)$ to $(m,n)$. In this paper, we give a formula to calculate $C(m,n)$ for arbitrary $m$ and $n$.
Koroljuk gave a summation formula for counting the number of lattice paths from $(0,0)$ to $(m,n)$ with $(1,0), (0,1)$-steps in the plane that stay strictly above the line $y=k(x-d)$, where $k$ and $d$ are positive integers. In this paper we obtain an explicit formula for the number of lattice paths from $(a,b)$ to $(m,n)$ above the diagonal $y=kx-r$, where $r$ is a rational number. Our result slightly generalizes Koroljuk's formula, while the former can be essentially derived from the latter. However, our proof uses a recurrence with respect to the starting points, and hereby presents a new a…
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 $β$-Hermite and $β$-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.
We solve two problems regarding the enumeration of lattice paths in $\mathbb{Z}^2$ with steps $(1,1)$ and $(1,-1)$ with respect to the major index, defined as the sum of the positions of the valleys, and to the number of certain crossings. The first problem considers crossings of a single path with a fixed horizontal line. The second one counts pairs of paths with respect to the number of times they cross each other. Our proofs introduce lattice path bijections with convenient visual descriptions, and the answers are given by remarkably simple formulas involving $q$-binomial coefficients.
The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a,b,c,d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindström-Gessel-Viennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates the numbers of configurations with b=d.
A circular Pascal array is a periodization of the familiar Pascal's triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain lattice paths within corridors, which are related to Dyck paths. This link provides new, short proofs of some nontrivial formulas found in the lattice-path literature.
We point out that the total number of trails and the total number of paths of given length, between two vertices of a simple undirected graph, are obtained as expectation values of specifically engineered quantum mechanical observables. Such observables are contextual with some background independent theories of gravity and emergent geometry. Thus, we point out yet another situation in which the mathematical formalism of a physical theory has some computational aspects involving intractable problems.
The great enumerator Germain Kreweras empirically discovered this intriguing fact, and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators, for example, Heinrich Niederhausen[N], Ira Gessel[G1], and Mireille Bousquet-Mélou[B], found other ingenious, ``simpler'' proofs. Yet none of them is as simple as ours! Our proof (with the generous help of our faithful computers) is ``ugly'' in the traditional sense, since it would be painful for a lowly human to follow all the steps. But according to our humble aesthetic taste, this proof is much more elegant, s…
We present some exact expressions for the number of paths of a given length in a perfect $m$-ary tree. We first count the paths in perfect rooted $m$-ary trees and then use the results to determine the number of paths in perfect unrooted $m$-ary trees, extending a known result for binary trees.
We study the gaps $Δp_n$ between consecutive rank sizes in $r$-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller's result that $Δp_n \geq 1$, which resolved a longstanding conjecture of Stanley, by showing that $Δp_n \geq 2r$. We also obtain stronger bounds in the case that the poset has many substructures called threads.
We present refined enumeration formulas for lattice paths in $\mathbb{Z}^2$ with two kinds of steps, by keeping track of the number of descents (i.e., turns in a given direction), the major index (i.e., the sum of the positions of the descents), and the number of crossings. One formula considers crossings between a path and a fixed line; the other considers crossings between two paths. Building on the first paper of the series, which used lattice path bijections to give the enumeration with respect to major index and crossings, we obtain a refinement that keeps track of the number of descents.…
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…
It is known that the Hilbert space dimensionality for quasiparticles in an SU(2)_k Chern-Simons-Witten theory is given by the number of directed paths in certain Bratteli diagrams. We present an explicit formula for these numbers for arbitrary k. This is on the basis of a relation with Dyck paths and Chebyshev polynomials.
Say a digraph is k-free if it has no directed cycles of length at most k, for positive integers k. Thomasse conjectured that the number of induced 3-vertex directed paths in a simple 2-free digraph on n vertices is at most (n-1)n(n+1)/15. We present an unpublished result of Bondy proving that there are at most 2n^3/25 such paths, and prove that for the class of circular interval digraphs, an upper bound of n^3/16 holds. We also study the problem of bounding the number of (non-induced) 4-vertex paths in 3-free digraphs. We show an upper bound of 4n^4/75 using Bondy's result for Thomasse's conje…
Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles and paths, respectively. They satisfy recurrence relations with respect to elementary edge or vertex operations. They are related to other polynomials and can also be generalized to the bivariate cycle polynomial, the bivariate path polynomial and the trivariate cycle-path polynomial. And a most general digraph polynomial satisfying such a linear recurrence …
We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat $F=(1,0)$ steps. We say that a Motzkin path of length $n$ is symmetric if its reflection about the line $x=n/2$ is itself. In this paper, we show that the number of self-conjugate $(s,s+1,s+2)$-cores is equal to the number of symmetric Motzkin paths of length $s$, and give a closed formula for this number.
The purpose of this article is to provide a solution to the $m$-fold Laplace equation in the half space $R_+^d$ under certain Dirichlet conditions. The solutions we present are a series of $m$ boundary layer potentials. We give explicit formulas for these layer potentials as linear combinations of powers of the Laplacian applied to the Dirichlet data, with coefficients determined by certain path counting problems.
A weighted coloured-edge graph is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such a graph paths are compared by their total weight in each colour, resulting in a Pareto set of minimal paths from one vertex to another. This paper will give a tight upper bound on the cardinality of a minimal set of paths for any weighted coloured-edge graph. Additionally, a bound is presented on the expected number of minimal paths in weighted bicoloured-ed…
