Lattice Path Enumeration Research Articles

Pricing Parisian Options by Generating Functions

We provide new and discrete time combinatorial approaches to evaluate Parisian options. By using generating functions, a very useful tool in lattice path enumeration, the computation of consecutive and cumulative Parisian options is significantly simplified. This leads to combinatorial algorithms that reduce the computational complexity and produce highly accurate prices.

Exact solutions of lattice polymer models

We consider directed path models of a selection of polymer and vesicle problems. Each model is used to illustrate an important method of solving lattice path enumeration problems. In particular, the Temperley method is used for the polymer collapse problem. The ZL method is used to solve the semi-continuous vesicle model. The Constant Term method is used to solve a set of partial difference equations for the polymer adsorption problem. The Kernel method is used to solve the functional equation that arises in the polymer force problem. Finally, the Transfer Matrix method is used to solve a problem in colloid dispersions. All these methods are combinatorially similar as they all construct equations by considering the action of adding an additional column to the set of objects.

Partially directed paths in a wedge

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional equation for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by Y = ± p X , and an asymmetric wedge defined by the lines Y = p X and Y = 0 , where p > 0 is an integer. We prove that the growth constant for all these models is equal to 1 + 2 , independent of the angle of the wedge. We derive function equations for both models, and obtain explicit expressions for the generating functions when p = 1 . From these we find asymptotic formulas for the number of partially directed paths of length n in a wedge when p = 1 . The functional equations are solved by a variation of the kernel method, which we call the “iterated kernel method.” This method appears to be similar to the obstinate kernel method used by Bousquet-Mélou (see, for example, references [M. Bousquet-Mélou, Counting walks in the quarter plane, in: Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, Trends Math., Birkhäuser, 2002, pp. 49–67; M. Bousquet-Mélou, Four classes of pattern-avoiding permutations under one roof: Generating trees with two labels, Electron. J. Combin. 9 (2) (2003) R19; M. Bousquet-Mélou, M. Petkovšek, Walks confined in a quadrant are not always D-finite, Theoret. Comput. Sci. 307 (2) (2003) 257–276]). This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi θ-functions, and have natural boundaries on the unit circle.

Matchings Avoiding Partial Patterns and Lattice Paths

In this paper, we consider matchings avoiding partial patterns $1123$ and $1132$. We give a bijection between $1123$-avoiding matchings with $n$ edges and nonnegative lattice paths from $(0,2)$ to $(2n,0)$. As a consequence, the refined enumeration of $1123$-avoiding matchings can be reduced to the enumeration of certain lattice paths. Another result of this paper is a bijection between $1132$-avoiding matchings with $n$ edges and lattice paths from $(0,0)$ to $(2n,0)$ starting with an up step, which may go under the $x$-axis.

Elliptic enumeration of nonintersecting lattice paths

We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elliptic generalization of the binomial coefficient. Convolution gives an identity equivalent to Frenkel and Turaev's V 9 10 summation. This appears to be the first combinatorial proof of the latter, and at the same time of some important degenerate cases including Jackson's ϕ 7 8 and Dougall's F 6 7 summation. By considering nonintersecting lattice paths we are led to a multivariate extension of the V 9 10 summation which turns out to be a special case of an identity originally conjectured by Warnaar, later proved by Rosengren. We conclude with discussing some future perspectives.

The Flagged Cauchy Determinant

We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing different starting and end points for the lattice paths, we are led to a lattice path proof of an identity of Gessel which expresses a Cauchy-like sum of Schur functions in terms of the complete symmetric functions.

A type-B associahedron

The (type-A) associahedron is a polytope related to polygon dissections which arises in several mathematical subjects. We propose a B-analogue of the associahedron. Our original motivation was to extend the analogies between type-A and type-B noncrossing partitions, by exhibiting a simplicial polytope whose h-vector is given by the rank-sizes of the type-B noncrossing partition lattice, just as the h-vector of the (simplicial type-A) associahedron is given by the Narayana numbers. The desired polytope Q B n is constructed via stellar subdivisions of a simplex, similarly to Lee's construction of the associahedron. As in the case of the (type-A) associahedron, the faces of Q B n can be described in terms of dissections of a convex polygon, and the f-vector can be computed from lattice path enumeration. Properties of the simple dual Q B∗ n are also discussed and the construction of a space tessellated by Q B∗ n is given. Additional analogies and relations with type A and further questions are also discussed.

Recursive initial value problems for Sheffer sequences

We consider certain systems of linear operator equations where the nth-order solution is made unique by an initial value taken from the preceding solution. Such conditions arise in the enumeration of lattice paths with weighted left turns, if the paths are restricted by a lower linear barrier of integer slope.

Combinatorial Statistics on Alternating Permutations

We consider two combinatorial statistics on permutations. One is the genus. The other, \widehat{\rm des}, is defined for alternating permutations, as the sum of the number of descents in the subwords formed by the peaks and the valleys. We investigate the distribution of des on genus zero permutations and Baxter permutations. Our q-enumerative results relate the des statistic to lattice path enumeration, the rank generating function and characteristic polynomial of noncrossing partition lattices, and polytopes obtained as face-figures of the associahedron.

Symmetric Sheffer sequences and their applications to lattice path counting

A sequence of Sheffer polynomials is symmetric, if the value of the nth degree polynomial at any natural number m agrees with the mth degree polynomial at n. While the above property sounds rather esoteric, symmetric Sheffer sequences frequently describe the elegant results of standard lattice path enumeration. We characterize all symmetric Sheffer sequences, and explain their role from the initial value problem point of view. Applications occur in the enumeration of nonintersecting weighted lattice paths, and the discussion of certain correlated random walks.

Enumeration of restricted three-dimensional lattice paths with fixed numbers of turns and an application

Consider three-dimensional lattice paths from the origin to ( rn + a, n, m), not touching the plane x = ry + a and having fixed numbers of turns. In this note, such paths are enumerated and an application is given.

Extension of certain types of generating relations

This paper gives a new expansion formula which essentially involves a double sum. As a consequence of our main result, eq. (6), various types of generating relations are seen to emerge and relevance with some known results is pointed out briefly. The usefulness of our main result is also indicated by considering its application to a probabilistic method for a lattice path enumeration problem.

Enumeration of lattice paths and generating functions for skew plane partitions

n-dimensional lattice paths not touching the hyperplanesX i−X i+1=−1,i=1,2,...,n, are counted by four different statistics, one of which is MacMahon's major index. By a reflection-like proof, heavily relying on Zeilberger's (Discrete Math. 44(1983), 325–326) solution of then-candidate ballot problem, determinantal expressions are obtained. As corollaries the generating functions for skew plane partitions, column-strict skew plane partitions, reverse skew plane plane partitions and column-strict reverse skew plane partitions, respectively, are evaluated, thus establishing partly new results, partly new proofs for known theorems in the theory of plane partitions.

A probabilistic method for lattice path enumeration

A probabilistic method for obtaining functional equations for lattice path enumeration problems is studied.

A factorization for formal laurent series and lattice path enumeration

If f = Σ n=−∞ ∞ a n t n is a formal Laurent series with certain restrictions on the a n , then f = f − f 0 f +, where f − contains only negative powers of t, f + contains only positive powers of t, and f 0 is independent of t. Applications include Lagrange's formula for series reversion, the problem of counting lattice paths below a diagonal, and a theorem of Furstenberg that the diagonal of a rational power series in two variables is algebraic.

Enumeration of higher-dimensional paths under restrictions

The enumeration of lattice paths lying between two boundaries in two dimensional space has been done and the explicit expression is a determinant. By considering a natural extension of the boundaries in higher dimensional space, a generalized recurrence relation is established, the solution of which gives the number of paths in higher dimension not crossing the boundaries, in a determinant form.

The enumeration of some restricted matrices

Some enumeration problems involving certain restricted matrices considered by L. Carlitz are reduced to lattice path enumeration problems which are then solved by means of first and last passage decomposition methods.

On Lattice Paths with Several Diagonal Steps

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.