Counting partially directed walks in a symmetric wedge
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| Description: |
Prellberg, T (QMUL)
Friday 11 April 2008, 14:15-15:00 Combinatorial Identities and their Applications in Statistical Mechanics |
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| Created: | 2008-04-29 11:26 | ||
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| Collection: | Combinatorics and Statistical Mechanics | ||
| Publisher: | Isaac Newton Institute | ||
| Copyright: | Prellberg, T | ||
| Language: | eng (English) | ||
| Distribution: |
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| Explicit content: | No | ||
| Aspect Ratio: | 4:3 | ||
| Screencast: | No | ||
| Bumper: | /sms-ingest/static/new-4x3-bumper.dv | ||
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| Abstract: | 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) |
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