130 research outputs found
Enumeration of m-ary cacti
The purpose of this paper is to enumerate various classes of cyclically
colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is
motivated by the topological classification of complex polynomials having at
most m critical values, studied by Zvonkin and others. We obtain explicit
formulae for both labelled and unlabelled m-ary cacti, according to i) the
number of polygons, ii) the vertex-color distribution, iii) the vertex-degree
distribution of each color. We also enumerate m-ary cacti according to the
order of their automorphism group. Using a generalization of Otter's formula,
we express the species of m-ary cacti in terms of rooted and of pointed cacti.
A variant of the m-dimensional Lagrange inversion is then used to enumerate
these structures. The method of Liskovets for the enumeration of unrooted
planar maps can also be adapted to m-ary cacti.Comment: LaTeX2e, 28 pages, 9 figures (eps), 3 table
On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution
International audienc
Many faces of symmetric edge polytopes
Symmetric edge polytopes are a class of lattice polytopes constructed from
finite simple graphs. In the present paper we highlight their connections to
the Kuramoto synchronization model in physics -- where they are called
adjacency polytopes -- and to Kantorovich--Rubinstein polytopes from finite
metric space theory. Each of these connections motivates the study of symmetric
edge polytopes of particular classes of graphs. We focus on such classes and
apply algebraic-combinatorial methods to investigate invariants of the
associated symmetric edge polytopes.Comment: 35 pages, 8 figures. Comments are very welcome
Lots and Lots of Perrin-Type Primality Tests and Their Pseudo-Primes
We use Experimental Mathematics and Symbolic Computation (with Maple), to
search for lots and lots of Perrin- and Lucas- style primality tests, and try
to sort the wheat from the chaff. More impressively, we find quite a few such
primality tests for which we can explicitly construct infinite families of
pseudo-primes, rather, like in the cases of Perrin pseudo-primes and the famous
Carmichael primes, only proving the mere existence of infinitely many of them.Comment: 9 pages. Accompanied by a Maple package and numerous output files
from <A
HREF="https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/perrin.html">this
url. This version corrects two typos pointed out by OEIS editor Hugo
Pfoertner, and lists the 7 smallest pseudo-primes found by Manuel Kauers for
the new primality test in the pape
Gibbs and Quantum Discrete Spaces
Gibbs measure is one of the central objects of the modern probability,
mathematical statistical physics and euclidean quantum field theory. Here we
define and study its natural generalization for the case when the space, where
the random field is defined is itself random. Moreover, this randomness is not
given apriori and independently of the configuration, but rather they depend on
each other, and both are given by Gibbs procedure; We call the resulting object
a Gibbs family because it parametrizes Gibbs fields on different graphs in the
support of the distribution. We study also quantum (KMS) analog of Gibbs
families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure
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