1,751 research outputs found
Enumeration of simple random walks and tridiagonal matrices
We present some old and new results in the enumeration of random walks in one
dimension, mostly developed in works of enumerative combinatorics. The relation
between the trace of the -th power of a tridiagonal matrix and the
enumeration of weighted paths of steps allows an easier combinatorial
enumeration of the paths. It also seems promising for the theory of tridiagonal
random matrices .Comment: several ref.and comments added, misprints correcte
Sequences of labeled trees related to Gelfand-Tsetlin patterns
By rewriting the famous hook-content formula it easily follows that there are
semistandard
tableaux of shape with entries in or,
equivalently, Gelfand-Tsetlin patterns with bottom row . In this
article we introduce certain sequences of labeled trees, the signed enumeration
of which is also given by this formula. In these trees, vertices as well as
edges are labeled, the crucial condition being that each edge label lies
between the vertex labels of the two endpoints of the edge. This notion enables
us to give combinatorial explanations of the shifted antisymmetry of the
formula and its polynomiality. Furthermore, we propose to develop an analog
approach of combinatorial reasoning for monotone triangles and explain how this
may lead to a combinatorial understanding of the alternating sign matrix
theorem.Comment: 26 pages, 12 figure
Parking functions, labeled trees and DCJ sorting scenarios
In genome rearrangement theory, one of the elusive questions raised in recent
years is the enumeration of rearrangement scenarios between two genomes. This
problem is related to the uniform generation of rearrangement scenarios, and
the derivation of tests of statistical significance of the properties of these
scenarios. Here we give an exact formula for the number of double-cut-and-join
(DCJ) rearrangement scenarios of co-tailed genomes. We also construct effective
bijections between the set of scenarios that sort a cycle and well studied
combinatorial objects such as parking functions and labeled trees.Comment: 12 pages, 3 figure
Solving multivariate functional equations
This paper presents a new method to solve functional equations of
multivariate generating functions, such as
giving a
formula for in terms of a sum over finite sequences. We use this
method to show how one would calculate the coefficients of the generating
function for parallelogram polyominoes, which is impractical using other
methods. We also apply this method to answer a question from fully commutative
affine permutations.Comment: 11 pages, 1 figure. v3: Main theorems and writing style revised for
greater clarity. Updated to final version, to appear in Discrete Mathematic
Continued fractions for permutation statistics
We explore a bijection between permutations and colored Motzkin paths that
has been used in different forms by Foata and Zeilberger, Biane, and Corteel.
By giving a visual representation of this bijection in terms of so-called cycle
diagrams, we find simple translations of some statistics on permutations (and
subsets of permutations) into statistics on colored Motzkin paths, which are
amenable to the use of continued fractions. We obtain new enumeration formulas
for subsets of permutations with respect to fixed points, excedances, double
excedances, cycles, and inversions. In particular, we prove that cyclic
permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
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