13,142 research outputs found
A new transfer-matrix algorithm for exact enumerations: Self-avoiding polygons on the square lattice
We present a new and more efficient implementation of transfer-matrix methods
for exact enumerations of lattice objects. The new method is illustrated by an
application to the enumeration of self-avoiding polygons on the square lattice.
A detailed comparison with the previous best algorithm shows significant
improvement in the running time of the algorithm. The new algorithm is used to
extend the enumeration of polygons to length 130 from the previous record of
110.Comment: 17 pages, 8 figures, IoP style file
Enumeration of self-avoiding walks on the square lattice
We describe a new algorithm for the enumeration of self-avoiding walks on the
square lattice. Using up to 128 processors on a HP Alpha server cluster we have
enumerated the number of self-avoiding walks on the square lattice to length
71. Series for the metric properties of mean-square end-to-end distance,
mean-square radius of gyration and mean-square distance of monomers from the
end points have been derived to length 59. Analysis of the resulting series
yields accurate estimates of the critical exponents and
confirming predictions of their exact values. Likewise we obtain accurate
amplitude estimates yielding precise values for certain universal amplitude
combinations. Finally we report on an analysis giving compelling evidence that
the leading non-analytic correction-to-scaling exponent .Comment: 24 pages, 6 figure
Self-avoiding walks and polygons on the triangular lattice
We use new algorithms, based on the finite lattice method of series
expansion, to extend the enumeration of self-avoiding walks and polygons on the
triangular lattice to length 40 and 60, respectively. For self-avoiding walks
to length 40 we also calculate series for the metric properties of mean-square
end-to-end distance, mean-square radius of gyration and the mean-square
distance of a monomer from the end points. For self-avoiding polygons to length
58 we calculate series for the mean-square radius of gyration and the first 10
moments of the area. Analysis of the series yields accurate estimates for the
connective constant of triangular self-avoiding walks, ,
and confirms to a high degree of accuracy several theoretical predictions for
universal critical exponents and amplitude combinations.Comment: 24 pages, 6 figure
Motzkin Intervals and Valid Hook Configurations
We define a new natural partial order on Motzkin paths that serves as an
intermediate step between two previously-studied partial orders. We provide a
bijection between valid hook configurations of -avoiding permutations and
intervals in these new posets. We also show that valid hook configurations of
permutations avoiding (or equivalently, ) are counted by the same
numbers that count intervals in the Motzkin-Tamari posets that Fang recently
introduced, and we give an asymptotic formula for these numbers. We then
proceed to enumerate valid hook configurations of permutations avoiding other
collections of patterns. We also provide enumerative conjectures, one of which
links valid hook configurations of -avoiding permutations, intervals in
the new posets we have defined, and certain closed lattice walks with small
steps that are confined to a quarter plane.Comment: 22 pages, 8 figure
On the weighted enumeration of alternating sign matrices and descending plane partitions
We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices
and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359]
that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs)
for which the 1 of the first row is in column k+1 and there are exactly m -1's
and m+p inversions is equal to the number of descending plane partitions (DPPs)
for which each part is at most n and there are exactly k parts equal to n, m
special parts and p nonspecial parts. The proof involves expressing the
associated generating functions for ASMs and DPPs with fixed n as determinants
of nxn matrices, and using elementary transformations to show that these
determinants are equal. The determinants themselves are obtained by standard
methods: for ASMs this involves using the Izergin-Korepin formula for the
partition function of the six-vertex model with domain-wall boundary
conditions, together with a bijection between ASMs and configurations of this
model, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem,
together with a bijection between DPPs and certain sets of nonintersecting
lattice paths.Comment: v2: published versio
Enumerations of lattice animals and trees
We have developed an improved algorithm that allows us to enumerate the
number of site animals on the square lattice up to size 46. We also calculate
the number of lattice trees up to size 44 and the radius of gyration of both
lattice animals and trees up to size 42. Analysis of the resulting series
yields an improved estimate, , for the growth constant
of lattice animals, and, , for the growth constant of
trees, and confirms to a very high degree of certainty that both the animal and
tree generating functions have a logarithmic divergence. Analysis of the radius
of gyration series yields the estimate, , for the size
exponent.Comment: 14 pages, 2 eps figures, corrections to some series coefficients and
reference
A parallel algorithm for the enumeration of benzenoid hydrocarbons
We present an improved parallel algorithm for the enumeration of fixed
benzenoids B_h containing h hexagonal cells. We can thus extend the enumeration
of B_h from the previous best h=35 up to h=50. Analysis of the associated
generating function confirms to a very high degree of certainty that and we estimate that the growth constant and the amplitude .Comment: 14 pages, 6 figure
A doubly-refined enumeration of alternating sign matrices and descending plane partitions
It was shown recently by the authors that, for any n, there is equality
between the distributions of certain triplets of statistics on nxn alternating
sign matrices (ASMs) and descending plane partitions (DPPs) with each part at
most n. The statistics for an ASM A are the number of generalized inversions in
A, the number of -1's in A and the number of 0's to the left of the 1 in the
first row of A, and the respective statistics for a DPP D are the number of
nonspecial parts in D, the number of special parts in D and the number of n's
in D. Here, the result is generalized to include a fourth statistic for each
type of object, where this is the number of 0's to the right of the 1 in the
last row of an ASM, and the number of (n-1)'s plus the number of rows of length
n-1 in a DPP. This generalization is proved using the known equality of the
three-statistic generating functions, together with relations which express
each four-statistic generating function in terms of its three-statistic
counterpart. These relations are obtained by applying the Desnanot-Jacobi
identity to determinantal expressions for the generating functions, where the
determinants arise from standard methods involving the six-vertex model with
domain-wall boundary conditions for ASMs, and nonintersecting lattice paths for
DPPs.Comment: 28 pages; v2: published versio
- …