Iwan Jensen and Andrew Rechnitzer 2008 J. Phys. A: Math. Theor. 41 215002 doi:10.1088/1751-8113/41/21/215002
The exact perimeter generating function for a model of punctured staircase polygons
Iwan Jensen1 and Andrew Rechnitzer2
Show affiliationsWe have derived the perimeter generating function of a model of punctured staircase polygons in which the internal staircase polygon is rotated by a 90° angle with respect to the outer staircase polygon. In one approach we calculated a long series expansion for the problem and found that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 4. We then solved this ODE and found a closed form expression for the generating function. This is a highly unusual and most fortuitous result since ODEs of such high order very rarely permit a closed form solution. In a second approach we proved the result for the generating function exactly using combinatorial arguments. This latter solution allows many generalizations including models with other types of punctures and a model with any fixed number of nested rotated staircase punctures.
- PACS
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.30.Lt Sequences, series, and summability
- MSC
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82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
30F35 Fuchsian groups and automorphic functions (See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx)
- Subjects
- Dates
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Issue 21 (30 May 2008)
Received 4 March 2008
Published 6 May 2008
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The exact perimeter generating function for a model of punctured staircase polygons
Iwan Jensen and Andrew Rechnitzer 2008 J. Phys. A: Math. Theor. 41 215002
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