Weighted lattice paths

Fray, Robert; Roselle, David P.

doi:10.2140/pjm.1971.37.85

Cited by 28 publications

(5 citation statements)

References 10 publications

(9 reference statements)

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“…On the square lattice, denoted S, many problems concerning one and more directed walks have been solved exactly, and so this task has essentially been accomplished. [10][11][12][13][14][15] However, a less complete set of problems have also been solved on the triangular lattice, denoted T. The literature pertinent to T directed lattice walks is still fairly large; [16][17][18][19][20][21][22][23][24][25][26] more references are listed in Refs. 27 and 28.…”

Section: Introductionmentioning

confidence: 99%

Anisotropic Step, Surface Contact, and Area Weighted Directed Walks on the Triangular Lattice

Oppenheim

Brak

Owczarek

2002

Int. J. Mod. Phys. B

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We present results for the generating functions of single fully-directed walks on the triangular lattice, enumerated according to each type of step and weighted proportional to the area between the walk and the surface of a half-plane (wall), and the number of contacts made with the wall. We also give explicit formulae for total area generating functions, that is when the area is summed over all configurations with a given perimeter, and the generating function of the moments of heights above the wall (the first of which is the total area). These results generalise and summarise nearly all known results on the square lattice: all the square lattice results can be obtaining by setting one of the step weights to zero. Our results also contain as special cases those that already exist for the triangular lattice. In deriving some of the new results we utilise the Enumerating Combinatorial Objects (ECO) and marked area methods of combinatorics for obtaining functional equations in the most general cases. In several cases we give our results both in terms of ratios of infinite q-series and as continued fractions.

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Section: Introductionmentioning

confidence: 99%

Anisotropic Step, Surface Contact, and Area Weighted Directed Walks on the Triangular Lattice

Oppenheim

Brak

Owczarek

2002

Int. J. Mod. Phys. B

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“…See [9] for the Lucas property of D x,y,z n,k . It is shown [2,4] that Proof. It suffices to show that {γ (n, k) := β(n + k + 1, k)} n,k∈N 0 satisfies (4.22) and (4.23) for (x, y, z) = (1, −1, 2).…”

Section: Hypergeometric Functions and Weighted Delannoy Numbersmentioning

confidence: 99%

“…Fray and Roselle [4] considered the recursion relation f (n, k) = xf (n − 1, k) + yf (n, k − 1) + zf (n − 1, k − 1) (n, k 1) This recurrence relation occurs in counting weighted lattice paths from (0, 0) to (n, k), where the horizontal steps are assigned the weight x, the vertical steps the weight y, and the diagonal steps the weight z. When x = y = z = 1 they reduce to the ordinary Delannoy numbers.…”

Section: Hypergeometric Functions and Weighted Delannoy Numbersmentioning

confidence: 99%

Integral polynomial sequences arising from matrix powers of order 2

Cheon

Lim

2013

Linear Algebra and its Applications

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In this paper we study a recursive system of integers {χ(n, k) : n > k 0};which is uniquely determined by the initial values {χ(n, 0)} ∞ n=1 . We show under the constant initial dates χ(n, 0) = χ(1, 0) for all n that the polynomial χ n (x) = n−1 k=0 χ(n, k)x k of degree n − 1 is (anti) palindromic. Several explicit formulae for χ(n, k) via Vandermonde matrix, mirrored -matrix, weighed Delannoy number, Riordan array, hypergeometric function, Jacobi polynomial, and some combinatorial identities are derived.

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“…with the initial values wt(D n,0 ) = x n and wt(D 0,k ) = y k , for n, k 0. The closed-form expression for the weighted Delannoy number is given by (see for example [20])…”

Section: Counting Families Of Non-intersecting Paths Connecting Four ...mentioning

confidence: 99%

An Extension of the Lindström-Gessel-Viennot Theorem

Lee

2022

Electron. J. Combin.

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Consider a weighted directed acyclic graph $G$ having an upward planar drawing. We give a formula for the total weight of the families of non-intersecting paths on $G$ with any given starting and ending points. While the Lindström-Gessel-Viennot theorem gives the signed enumeration of these weights (according to the connection type), our result provides the straight count, expressing it as a determinant whose entries are signed counts of lattice paths with given starting and ending points.

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Weighted lattice paths

Cited by 28 publications

References 10 publications

Anisotropic Step, Surface Contact, and Area Weighted Directed Walks on the Triangular Lattice

Anisotropic Step, Surface Contact, and Area Weighted Directed Walks on the Triangular Lattice

Integral polynomial sequences arising from matrix powers of order 2

An Extension of the Lindström-Gessel-Viennot Theorem

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