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Walks confined in a quadrant are not always Dfinite
"... We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z , and always stay in the quadrant x 0; y 0. We first give a criterion which guarantees that the length generating function of these walks is Dfinite, that is, satisfies a li ..."
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We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z , and always stay in the quadrant x 0; y 0. We first give a criterion which guarantees that the length generating function of these walks is Dfinite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion
Partially directed paths in a wedge
 Journal of Combinatorial Theory, Series A
"... 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 recurrence for the generating function, and solving for it. In this paper we consider a mo ..."
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Cited by 6 (0 self)
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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 recurrence 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 = ±pX, and an asymmetric wedge defined by the lines Y = pX 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 functional recursions 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 recurrences 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 BousquetMélou (see, for example, references [5, 6]). 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 θfunctions, and have natural boundaries on the unit circle.
On the Enumeration of Noncrossing Pairings of Wellbalanced Binary Strings
"... A noncrossingpairingon abinarystringpairsones andzeroes such that thearcs representing the pairings are noncrossing. A binary string is wellbalanced if it is of the form 1 a1 0 a1 ..."
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A noncrossingpairingon abinarystringpairsones andzeroes such that thearcs representing the pairings are noncrossing. A binary string is wellbalanced if it is of the form 1 a1 0 a1