159 research outputs found
Walks on the slit plane: other approaches
Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is
a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i
belongs to S for all i, and none of the points w_i, i>0, lie on the half-line
H= {(k,0): k =< 0}.
In a recent paper, G. Schaeffer and the author computed the length generating
function S(t) of walks on the slit plane for several sets S. All the generating
functions thus obtained turned out to be algebraic: for instance, on the
ordinary square lattice,
S(t) =\frac{(1+\sqrt{1+4t})^{1/2}(1+\sqrt{1-4t})^{1/2}}{2(1-4t)^{3/4}}.
The combinatorial reasons for this algebraicity remain obscure.
In this paper, we present two new approaches for solving slit plane models.
One of them simplifies and extends the functional equation approach of the
original paper. The other one is inspired by an argument of Lawler; it is more
combinatorial, and explains the algebraicity of the product of three series
related to the model. It can also be seen as an extension of the classical
cycle lemma. Both methods work for any set of steps S.
We exhibit a large family of sets S for which the generating function of
walks on the slit plane is algebraic, and another family for which it is
neither algebraic, nor even D-finite. These examples give a hint at where the
border between algebraicity and transcendence lies, and calls for a complete
classification of the sets S.Comment: 31 page
Walks confined in a quadrant are not always D-finite
We consider planar lattice walks that start from a prescribed position, take
their steps in a given finite subset of Z^2, and always stay in the quadrant x
>= 0, y >= 0. We first give a criterion which guarantees that the length
generating function of these walks is D-finite, that is, satisfies a linear
differential equation with polynomial coefficients. This criterion applies,
among others, to the ordinary square lattice walks. Then, we prove that walks
that start from (1,1), take their steps in {(2,-1), (-1,2)} and stay in the
first quadrant have a non-D-finite generating function. Our proof relies on a
functional equation satisfied by this generating function, and on elementary
complex analysis.Comment: To appear in Theoret. Comput. Sci. (special issue devoted to random
generation of combinatorial objects and bijective combinatorics
Decreasing subsequences in permutations and Wilf equivalence for involutions
In a recent paper, Backelin, West and Xin describe a map that
recursively replaces all occurrences of the pattern in a permutation
by occurrences of the pattern . The resulting
permutation contains no decreasing subsequence of length .
We prove that, rather unexpectedly, the map commutes with taking the
inverse of a permutation. In the BWX paper, the definition of is
actually extended to full rook placements on a Ferrers board (the permutations
correspond to square boards), and the construction of the map is the
key step in proving the following result. Let be a set of patterns starting
with the prefix . Let be the set of patterns obtained by
replacing this prefix by in every pattern of . Then for all ,
the number of permutations of the symmetric group \Sn_n that avoid equals
the number of permutations of \Sn_n that avoid . Our commutation result,
generalized to Ferrers boards, implies that the number of {\em involutions} of
\Sn_n that avoid is equal to the number of involutions of \Sn_n
avoiding , as recently conjectured by Jaggard
XML Compression via DAGs
Unranked trees can be represented using their minimal dag (directed acyclic
graph). For XML this achieves high compression ratios due to their repetitive
mark up. Unranked trees are often represented through first child/next sibling
(fcns) encoded binary trees. We study the difference in size (= number of
edges) of minimal dag versus minimal dag of the fcns encoded binary tree. One
main finding is that the size of the dag of the binary tree can never be
smaller than the square root of the size of the minimal dag, and that there are
examples that match this bound. We introduce a new combined structure, the
hybrid dag, which is guaranteed to be smaller than (or equal in size to) both
dags. Interestingly, we find through experiments that last child/previous
sibling encodings are much better for XML compression via dags, than fcns
encodings. We determine the average sizes of unranked and binary dags over a
given set of labels (under uniform distribution) in terms of their exact
generating functions, and in terms of their asymptotical behavior.Comment: A short version of this paper appeared in the Proceedings of ICDT
201
The degree distribution in bipartite planar maps: applications to the Ising model
We characterize the generating function of bipartite planar maps counted
according to the degree distribution of their black and white vertices. This
result is applied to the solution of the hard particle and Ising models on
random planar lattices. We thus recover and extend some results previously
obtained by means of matrix integrals.
Proofs are purely combinatorial and rely on the idea that planar maps are
conjugacy classes of trees. In particular, these trees explain why the
solutions of the Ising and hard particle models on maps of bounded degree are
always algebraic.Comment: 32 pages, 15 figure
Generating functions for generating trees
Certain families of combinatorial objects admit recursive descriptions in
terms of generating trees: each node of the tree corresponds to an object, and
the branch leading to the node encodes the choices made in the construction of
the object. Generating trees lead to a fast computation of enumeration
sequences (sometimes, to explicit formulae as well) and provide efficient
random generation algorithms. We investigate the links between the structural
properties of the rewriting rules defining such trees and the rationality,
algebraicity, or transcendence of the corresponding generating function.Comment: This article corresponds, up to minor typo corrections, to the
article submitted to Discrete Mathematics (Elsevier) in Nov. 1999, and
published in its vol. 246(1-3), March 2002, pp. 29-5
Asymptotic Behavior of Inflated Lattice Polygons
We study the inflated phase of two dimensional lattice polygons with fixed
perimeter and variable area, associating a weight to a
polygon with area and bends. For convex and column-convex polygons, we
show that , where , and . The
constant is found to be the same for both types of polygons. We argue
that self-avoiding polygons should exhibit the same asymptotic behavior. For
self-avoiding polygons, our predictions are in good agreement with exact
enumeration data for J=0 and Monte Carlo simulations for . We also
study polygons where self-intersections are allowed, verifying numerically that
the asymptotic behavior described above continues to hold.Comment: 7 page
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