34 research outputs found
Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice
Hexagonal polyominoes are polyominoes on the honeycomb lattice. We enumerate
the symmetry classes of convex hexagonal polyominoes. Here convexity is to be
understood as convexity along the three main column directions. We deduce the
generating series of free (i.e. up to reflection and rotation) and of
asymmetric convex hexagonal polyominoes, according to area and half-perimeter.
We give explicit formulas or implicit functional equations for the generating
series, which are convenient for computer algebra.Comment: 21 pages, 16 figures, 2 tables. This is the full version of a paper
presented at the FPSAC Conference in Vancouver, Canada, June 28 -- July 2,
200
Combinatorics of diagonally convex directed polyominoes
AbstractA new bijection between the diagonally convex directed (dcd-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, target cells). For a part of these results we also give another proof, which is based on Raney's generalized lemma. Thanks to the fact that the diagonals of a dcd-polyomino can grow at most by one, the problem of q-enumeration of this object can be solved by an application of Gessel's q-analog of the Lagrange inversion formula
The asymptotic determinant of the discrete Laplacian
We compute the asymptotic determinant of the discrete Laplacian on a
simply-connected rectilinear region in R^2. As an application of this result,
we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.Comment: 36 pages, 4 figures, to appear in Acta Mathematic
The number of directed -convex polyominoes
We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed -convex polyominoes.We show it is a rational function and we study its asymptotic behavior
Compact convex sets of the plane and probability theory
The Gauss-Minkowski correspondence in states the existence of
a homeomorphism between the probability measures on such that
and the compact convex sets (CCS) of the plane
with perimeter~1. In this article, we bring out explicit formulas relating the
border of a CCS to its probability measure. As a consequence, we show that some
natural operations on CCS -- for example, the Minkowski sum -- have natural
translations in terms of probability measure operations, and reciprocally, the
convolution of measures translates into a new notion of convolution of CCS.
Additionally, we give a proof that a polygonal curve associated with a sample
of random variables (satisfying ) converges
to a CCS associated with at speed , a result much similar to
the convergence of the empirical process in statistics. Finally, we employ this
correspondence to present models of smooth random CCS and simulations