Counting partially directed walks in a symmetric 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 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.

Related Links

• http://www.maths.qmul.ac.uk/~tp/papers/pub057.pdf - Partially directed paths in a wedge (van Rensburg; Prellberg; Rechnitzer)
• http://arxiv.org/abs/0712.2804v3 - Nestings of Matchings and Permutations and North Steps in PDSAWs (Rubey)
• http://arxiv.org/abs/0803.4233v1 - A Bijection Between Partially Directed Paths in the Symmetric Wedge and Matchings (Poznanovik)