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
-
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.30.Lt Sequences, series, and summability
- MSC
-
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
-
Issue 21 (30 May 2008)
Received 4 March 2008
Published 6 May 2008
-
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
-
The nuclear star cluster of the Milky Way
Rainer Schödel et al 2008 J. Phys.: Conf. Ser. 131 012044
-
The interplay between surface–water and hydrogen bonding in a water adlayer on Pt(111) and Ag(111)
Luigi Delle Site et al 2007 J. Phys.: Condens. Matter 19 242101
-
Doping poly[2-methoxy-5-(2'-ethylhexyloxy)-1,4-phenylenevinylene] with PbSe nanoparticles or fullerenes
D K Chambers et al 2008 J. Phys.: Condens. Matter 20 382202
-
Dephasing by electron–electron interactions in a ballistic Mach–Zehnder interferometer
Clemens Neuenhahn and Florian Marquardt 2008 New J. Phys. 10 115018
-
Linking optical and infrared observations with gravitational wave sources through transient variability
C W Stubbs 2008 Class. Quantum Grav. 25 184033
-
Implementation of standard testbeds for numerical relativity
M C Babiuc et al 2008 Class. Quantum Grav. 25 125012
-
Interplay between electronic states and structure during Au faceting
F Schiller et al 2008 New J. Phys. 10 113017
-
Fermions in loop quantum cosmology and the role of parity
Martin Bojowald and Rupam Das 2008 Class. Quantum Grav. 25 195006
-
Semiconductor to metal transition in SWNTs caused by interaction with gold and platinum nanoparticles
Rakesh Voggu et al 2008 J. Phys.: Condens. Matter 20 215211
Users also read
What's this?- One-dimensional anyons with competing δ-function and derivative δ-function potentials
- Taming the Yukawa potential singularity: improved evaluation of bound states and resonance energies
- Hannay angle and geometric phase shifts under adiabatic parameter changes in classical dissipative systems
-symmetric models of scatteringRelated review articles
What's this?- The effective temperature
- Unconventional phase transitions in a constrained single polymer chain
- Discrete breathers and the anomalous decay of luminescence