We address the problem of counting walks by winding angle on the Kreweras lattice, an oriented version of the triangular lattice. Our method uses a new decomposition of the lattice, which allows us to write functional equations characterising a generating function of walks counted by length, endpoint and winding angle. We then solve these functional equations in terms of Jacobi theta functions. By using this result in conjunction with the reflection principle, we count walks confined to a cone of opening angle any multiple of $\fracπ{3}$, allowing us to extract asymptotic and algebraic information for these walks. Our method and results extend analogously to three other lattices, including the square lattice and triangular lattice. On the square lattice, most of our results were derived by Timothy Budd in 2017, so the current work can be seen as an extension of Budd's results to the three other lattices that we consider. Budd's method of deducing these results was very different, as it was based on an explicit eigenvalue decomposition of certain matrices counting paths in the lattice.
A method is described to count simple diagonal walks on $\mathbb{Z}^2$ with a fixed starting point and endpoint on one of the axes and a fixed winding angle around the origin. The method involves the decomposition of such walks into smaller pieces, the generating functions of which are encoded in a commuting set of Hilbert space operators. The general enumeration problem is then solved by obtaining an explicit eigenvalue decomposition of these operators involving elliptic functions. By further restricting the intermediate winding angles of the walks to some open interval, the method can be use…
Formulas are derived for counting walks in the Kronecker product of graphs, and the associated spectral distributions are obtained by the Mellin convolution of probability distributions. Two-dimensional restricted lattices admitting the Kronecker product structure are listed, and their spectral distributions are calculated in terms of elliptic integrals.
The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walk
The connective constant $μ(G)$ of a graph $G$ is the asymptotic growth rate of the number of self-avoiding walks on $G$ from a given starting vertex. We survey three aspects of the dependence of the connective constant on the underlying graph $G$. Firstly, when $G$ is cubic, we study the effect on $μ(G)$ of the Fisher transformation (that is, the replacement of vertices by triangles). Secondly, we discuss upper and lower bounds for $μ(G)$ when $G$ is regular. Thirdly, we present strict inequalities for the connective constants $μ(G)$ of vertex-transitive graphs $G$, as $G$ varies. As a consequ…
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 $(i,j) \in \mathbb{Z}^2_+$.
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 count the number of closed walks on a vertex in a regular tree using the Catalan's triangle and also the Borel's triangle, showing another combinatorial structure counted by these two array of numbers.
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.
We consider lattice walks in the plane starting at the origin, remaining in the first quadrant and made of West, South and North-East steps. In 1965, Germain Kreweras discovered a remarkably simple formula giving the number of these walks (with prescribed length and endpoint). Kreweras' proof was very involved and several alternative derivations have been proposed since then. But the elegant simplicity of the counting formula remained unexplained. We give the first purely combinatorial explanation of this formula. Our approach is based on a bijection between Kreweras walks and triangulations w…
The connective constant $μ(G)$ of a graph $G$ is the asymptotic growth rate of the number $σ_{n}$ of self-avoiding walks of length $n$ in $G$ from a given vertex. We prove a formula for the connective constant for free products of quasi-transitive graphs and show that $σ_{n}\sim A_{G} μ(G)^{n}$ for some constant $A_{G}$ that depends on $G$. In the case of finite products $μ(G)$ can be calculated explicitly and is shown to be an algebraic number.
For d at least two and integer n, let c_n = c_n(d) denote the number of length n self-avoiding walks beginning at the origin in the integer lattice Z^d, and, for even n, let p_n = p_n(d) denote the number of length n self-avoiding polygons in Z^d up to translation. Then the probability under the uniform law W_n on self-avoiding walks Gamma of any given odd length n beginning at the origin that Gamma closes -- i.e., that Gamma's endpoint is a neighbour of the origin -- is given by W_n ( Gamma closes ) = 2(n+1) p_{n+1}/c_n. The polygon and walk cardinalities share a common exponential growth: li…
We define numbers of the type Oj(N) and Ej(N) and the corresponding integer sequences. We prove that these integer sequences, e.g., SO(N) and SE(N) correspond to the number of odd and even walks in complete graphs. We then prove that there is a unique family of graphs which have exactly the same sequence of odd walks between connected nodes and of even walks between pairs of nodes at distance two, respectively. These graphs are obtained as the Kronecker product. We show that they are the incidence graphs of block designs, are distance-regular and Ramanujan graphs.
In this paper, we study a family of lattice walks which are related to the Hadamard conjecture. There is a bijection between paths of these walks which originate and terminate at the origin and equivalence classes of partial Hadamard matrices. Therefore, the existence of partial Hadamard matrices can be proved by showing that there is positive probability of a random walk returning to the origin after a specified number of steps. Moreover, the number of these designs can be approximated by estimating the return probabilities. We use the inversion formula for the Fourier transform of the random…
We present an average case analysis of two variants of dual-pivot quicksort, one with a non-algorithmic comparison-optimal partitioning strategy, the other with a closely related algorithmic strategy. For both we calculate the expected number of comparisons exactly as well as asymptotically, in particular, we provide exact expressions for the linear, logarithmic, and constant terms. An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proven.
We study walk algebras and Hecke algebras for Kac-Moody root systems. Each choice of orientation for the set of real roots gives rise to a corresponding "oriented" basis for each of these algebras. We show that the notion of distinguished subexpression naturally arises when studying the transition matrix between oriented bases. We then relate these notions to the geometry of Kac-Moody flag varieties and Bott-Samelson varieties. In particular, we show that the number of points over a finite field in certain intersections of these varieties is given by change of basis coefficients between orient…
Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of the numbers $q(i,j;n)$ of such walks starting at the origin and ending at $(i,j) \in {\bf Z}_+^{2}$ after $n$ steps is studied. For all non-singular models of walks, the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are continued as multi-valued functions on ${\bf C}$ having infinitely many meromorphic branches, of which the set of poles is identifi…
We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain of the generating functions are algebraic over the field of rational functions, while some others do not satisfy any algebraic differential equation with rational function coefficients. Our techniques involve differential Galois theory for difference equations as well as complex analysis (Weierstrass parameterization of elliptic curves). We also extend to th…
This is an exposition of the theorem from the title, which says that the number of self-avoiding walks with n steps in the hexagonal lattice has asymptotics (2cos(pi/8))^{n+o(n)}. We lift the key identity to formal level and simplify the part of the proof bounding the growth constant from below. In our calculation the lower bound comes from an identity asserting that a linear combination of 288 generating functions counting self-avoiding walks in a certain domain by length, final edge direction and winding number modulo 48 equals the geometric series 2cos(pi/8)x + (2cos(pi/8))^2x^2 + ... .
This paper is the first application of the compensation approach to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane $Z_{+}^{2}$ with a step set that is a subset of $\{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}$ in the interior of $Z_{+}^{2}$. We derive an explicit expression for the counting generating function, which turns out to be meromorphic and nonholonomic, can be easily inverted, and can be used to obtain asymptotic expressions for the counting coefficients.
We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0<q<1$. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polyn…
We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang-Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged u(n)-WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further t…
We approach the problem of counting the number of walks in a digraph from three different perspectives: enumerative combinatorics, linear algebra, and symbolic dynamics.
Eight combinatorial identities are listed and proved by counting paths in the one-dimensional random walk. Four of these identities are assumed to be new.
We propose an experimental mathematics approach leading to the computer-driven discovery of various structural properties of general counting functions coming from enumeration of walks.
We study planar walks that start from a given point (i\_0, j\_0), take their steps in a finite set S, and are confined in the first quadrant of the plane. Their enumeration can be attacked in a systematic way: the generating function Q(x, y, t) that counts them by their length (variable t) and the coordinates of their endpoint (variables x, y) satisfies a linear functional equation encoding the step-by-step description of walks. For instance, for the square lattice walks starting from the origin, this equation reads (xy-t(x + y + x^2 y + x y^2)) Q(x, y, t) = xy-xtQ(x, 0, t)-ytQ(O, y, t). The c…
Let $G$ be an infinite, vertex-transitive lattice with degree $λ$ and fix a vertex on it. Consider all cycles of length exactly $l$ from this vertex to itself on $G$. Erasing loops chronologically from these cycles, what is the fraction $F_p/λ^{\ell(p)}$ of cycles of length $l$ whose last erased loop is some chosen self-avoiding polygon $p$ of length $\ell(p)$, when $l\to\infty$ ? We use combinatorial sieves to prove an exact formula for $F_p/λ^{\ell(p)}$ that we evaluate explicitly. We further prove that for all self-avoiding polygons $p$, $F_p\in\mathbb{Q}[χ]$ with $χ$ an irrational number d…
On an $r\times (n-r)$ lattice rectangle, we first consider walks that begin at the SW corner, proceed with unit steps in either of the directions E or N, and terminate at the NE corner of the rectangle. For each integer $k$ we ask for $N_k^{n,r}$, the number of {\em ordered\/} pairs of these walks that intersect in exactly $k$ points. The number of points in the intersection of two such walks is defined as the cardinality of the intersection of their two sets of vertices, excluding the initial and terminal vertices. We find two explicit formulas for the numbers $N_k^{n,r}$. Next we note that $…
In the past fifteen years, the enumeration of lattice walks with steps takenin a prescribed set S and confined to a given cone, especially the firstquadrant of the plane, has been intensely studied. As a result, the generating functions ofquadrant walks are now well-understood, provided the allowed steps aresmall, that is $S \subset \{-1, 0,1\}^2$. In particular, having smallsteps is crucial for the definition of a certain group of bi-rationaltransformations of the plane. It has been proved that this group is finite ifand only if the corresponding generating function is D-finite (that is, it s…
We provide guessed recurrence equations for the counting sequences of rook paths on d-dimensional chess boards starting at (0..0) and ending at (n..n), where d=2,3,...,12. Our recurrences suggest refined asymptotic formulas of these sequences. Rigorous proofs of the guessed recurrences as well as the suggested asymptotic forms are posed as challenges to the reader.
This work develops a methodical approach to counting of walks on cartesian products, biproducts, symmetric and exterior powers and bipowers, Schur operations, coverings and semicoverings of weighted graphs. For weight and root lattices of semisimple Lie algebras, this approach allows us to compute various combinatorial and representation-theoretical constants, in particular, the number of plane symplectic wave graphs with given number of vertices.
We show that the method of counting closed walks in strongly regular graphs rules out no parameter sets other than those ruled out by the method of counting eigenvalue multiplicities.
In this article we obtain new expressions for the generating functions counting (non-singular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use solutions to boundary value problems. We illustrate our results with three examples (an algebraic case, a transcendental D-finite case, and an infinite group model).
We count the number of walks of length n on a k-node circular digraph that cover all k nodes in two ways. The first way illustrates the transfer-matrix method. The second involves counting various classes of height-restricted lattice paths. We observe that the results also count so-called k-balanced strings of length n, generalizing a 1996 Putnam problem.
We develop a theory of the field of double Laurent series, iterated Laurent series, and Malcev-Neumann series that applies to most constant term evaluation problems. These include (i) MacMahon's partition analysis, counting solutions of systems of linear Diophantine equations or inequalities, counting the number of lattice points in convex polytopes, (ii) evaluating combinatorial sums and their generating functions, and proving combinatorial identities, and (iii) lattice path enumeration such as walks on the slit plane and walks on the quarter plane. In the general setting of this new theory, …
