215,052 research outputs found
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
Generating functions for multi-labeled trees
Multi-labeled trees are a generalization of phylogenetic trees that are used, for example, in the study of gene versus species evolution and as the basis for phylogenetic network construction. Unlike phylogenetic trees, in a leaf-multi-labeled tree it is possible to label more than one leaf by the same element of the underlying label set. In this paper we derive formulae for generating functions of leaf-multi-labeled trees and use these to derive recursions for counting such trees. In particular,weprove results which generalize previous theorems by Harding on so-called tree-shapes, and by Otter on relating the number of rooted and unrooted phylogenetic trees
The Betti numbers of the moduli space of stable sheaves of rank 3 on P2
This article computes the generating functions of the Betti numbers of the
moduli space of stable sheaves of rank 3 on the projective plane P2 and its
blow-up. Wall-crossing is used to obtain the Betti numbers on the blow-up.
These can be derived equivalently using flow trees, which appear in the physics
of BPS-states. The Betti numbers for P2 follow from those for the blow-up by
the blow-up formula. The generating functions are expressed in terms of modular
functions and indefinite theta functions.Comment: 15 pages, final versio
Generating trees for permutations avoiding generalized patterns
We construct generating trees with one, two, and three labels for some
classes of permutations avoiding generalized patterns of length 3 and 4. These
trees are built by adding at each level an entry to the right end of the
permutation, which allows us to incorporate the adjacency condition about some
entries in an occurrence of a generalized pattern. We use these trees to find
functional equations for the generating functions enumerating these classes of
permutations with respect to different parameters. In several cases we solve
them using the kernel method and some ideas of Bousquet-M\'elou. We obtain
refinements of known enumerative results and find new ones.Comment: 17 pages, to appear in Ann. Com
Integrability of graph combinatorics via random walks and heaps of dimers
We investigate the integrability of the discrete non-linear equation
governing the dependence on geodesic distance of planar graphs with inner
vertices of even valences. This equation follows from a bijection between
graphs and blossom trees and is expressed in terms of generating functions for
random walks. We construct explicitly an infinite set of conserved quantities
for this equation, also involving suitable combinations of random walk
generating functions. The proof of their conservation, i.e. their eventual
independence on the geodesic distance, relies on the connection between random
walks and heaps of dimers. The values of the conserved quantities are
identified with generating functions for graphs with fixed numbers of external
legs. Alternative equivalent choices for the set of conserved quantities are
also discussed and some applications are presented.Comment: 38 pages, 15 figures, uses epsf, lanlmac and hyperbasic
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