# [seqfan] seq from enumerating Classical dimers on the triangular lattice

Jonathan Post jvospost3 at gmail.com
Thu Jan 7 20:37:44 CET 2010

```The following is adapated from
Classical dimers on the triangular lattice
Authors: P. Fendley, R. Moessner, S.L. Sondhi
http://arxiv.org/abs/cond-mat/0206159
[v2] May 14, 2003

p.5, TABLE I: Number of dimer coverings, Z, of triangular lattices
of with [sic] L_x × 2L_y sites

By antidiagonals:
1, 2, 1, 3, 5, 1, 5, 15, 13, 1, 8, 56, 85, 34, 1, 13, 203, 749, 493,
89, 1, 21, 749, 6475, 10293, 2871, 233, 1

The table begins:

.............L_y = 1.|.L_y = 2.|.L_y = 3.|.L_y = 4.|.L_y = 5.|.L_y =
6.|.L_y = 7.|
L_x = 1.|....1......|. ...1.... ..|.. ..1......|.. ..1......|..
..1......|.. ..1......|....1......|
L_x = 2.|....2......|.....5.... .|.. ..13.....|....34.....|..
..89.....|..233......|.
L_x = 3.|....3......|....15.... .|....85.....|...493.....|..2871.....|..16731.|.
L_x = 4.|....5......|....56.... .|...749.....|..10293..|
L_x = 5.|....8......|...203.... .|.. 6475..|
L_x = 6.|...13......|..749.... .|.
L_x = 7.|...21......|

The L_x=1 row is the all-ones A000012
The L_y=1 column seems to be the Fibonacci A000045, as does (except
for first value) the L_x=2 row.
The other rows and columns seem to me new to OEIS.

One has a related seq from the table, namely with periodic boundary
conditions, only those sizes
(L_x =>3 and L_y => 2) are given in which any pair of sites is
linked by at most one bond.

Abstract: We study the classical hard-core dimer model on the
triangular lattice. Following Kasteleyn's fundamental theorem on
planar graphs, this problem is soluble by Pfaffians. This model is
particularly interesting for, unlike the dimer problems on the
bipartite square and hexagonal lattices, its correlations are short
ranged with a correlation length of less than one lattice constant. We
compute the dimer-dimer and monomer-monomer correlators, and find that
the model is deconfining: the monomer-monomer correlator falls off
exponentially to a constant value sin(pi/12)/sqrt(3) = .1494..., only
slightly below the nearest-neighbor value of 1/6. We also consider the
anisotropic triangular lattice model in which the square lattice is
perturbed by diagonal bonds of one orientation and small fugacity. We
show that the model becomes non-critical immediately and that this
perturbation is equivalent to adding a mass term to each of two
Majorana fermions that are present in the long wavelength limit of the
square-lattice problem.

```