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Counting partially directed walks in a symmetric wedge

Friday 11th April 2008 - 14:15 to 15:00
INI Seminar Room 1

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 its solution.

We consider a model of partially directed walks from the origin in the square lattice confined to a symmetric wedge defined by Y = ±X.

We derive a functional equation for the generating function of the model, and obtain an explicit solution using a version of the Kernel method.

This solution shows that there is a direct connection with matchings of an 2n-set counted with respect to the number of crossings, and a bijective proof has since been obtained.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons