On FPL configurations with four sets of nested arches

Details On FPL configurations with four sets of nested arches On FPL configurations with four sets of nested arches

2004-03-10

arxiv.org/abs/cond-mat/0403268v2

Physics

Condensed Matter

Statistical Mechanics

22 pages, TeX, 16 figures; a new formula for a generating function added

Scientific paper

10.1088/1742-5468/2004/06/P06005

The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a,b,c,d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates the numbers of configurations with b=d.

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