415 research outputs found
On the distribution of surface extrema in several one- and two-dimensional random landscapes
We study here a standard next-nearest-neighbor (NNN) model of ballistic
growth on one- and two-dimensional substrates focusing our analysis on the
probability distribution function of the number of maximal points
(i.e., local ``peaks'') of growing surfaces. Our analysis is based on two
central results: (i) the proof (presented here) of the fact that uniform
one--dimensional ballistic growth process in the steady state can be mapped
onto ''rise-and-descent'' sequences in the ensemble of random permutation
matrices; and (ii) the fact, established in Ref. \cite{ov}, that different
characteristics of ``rise-and-descent'' patterns in random permutations can be
interpreted in terms of a certain continuous--space Hammersley--type process.
For one--dimensional system we compute exactly and also present
explicit results for the correlation function characterizing the enveloping
surface. For surfaces grown on 2d substrates, we pursue similar approach
considering the ensemble of permutation matrices with long--ranged
correlations. Determining exactly the first three cumulants of the
corresponding distribution function, we define it in the scaling limit using an
expansion in the Edgeworth series, and show that it converges to a Gaussian
function as .Comment: 25 pages, 12 figure
Commutative combinatorial Hopf algebras
We propose several constructions of commutative or cocommutative Hopf
algebras based on various combinatorial structures, and investigate the
relations between them. A commutative Hopf algebra of permutations is obtained
by a general construction based on graphs, and its non-commutative dual is
realized in three different ways, in particular as the Grossman-Larson algebra
of heap ordered trees.
Extensions to endofunctions, parking functions, set compositions, set
partitions, planar binary trees and rooted forests are discussed. Finally, we
introduce one-parameter families interpolating between different structures
constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245
Random patterns generated by random permutations of natural numbers
We survey recent results on some one- and two-dimensional patterns generated
by random permutations of natural numbers. In the first part, we discuss
properties of random walks, evolving on a one-dimensional regular lattice in
discrete time , whose moves to the right or to the left are induced by the
rise-and-descent sequence associated with a given random permutation. We
determine exactly the probability of finding the trajectory of such a
permutation-generated random walk at site at time , obtain the
probability measure of different excursions and define the asymptotic
distribution of the number of "U-turns" of the trajectories - permutation
"peaks" and "through". In the second part, we focus on some statistical
properties of surfaces obtained by randomly placing natural numbers on sites of a 1d or 2d square lattices containing sites. We
calculate the distribution function of the number of local "peaks" - sites the
number at which is larger than the numbers appearing at nearest-neighboring
sites - and discuss some surprising collective behavior emerging in this model.Comment: 16 pages, 5 figures; submitted to European Physical Journal,
proceedings of the conference "Stochastic and Complex Systems: New Trends and
Expectations" Santander, Spai
The Algebra of Binary Search Trees
We introduce a monoid structure on the set of binary search trees, by a
process very similar to the construction of the plactic monoid, the
Robinson-Schensted insertion being replaced by the binary search tree
insertion. This leads to a new construction of the algebra of Planar Binary
Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric
Functions and Free Symmetric Functions. We briefly explain how the main known
properties of the Loday-Ronco algebra can be described and proved with this
combinatorial point of view, and then discuss it from a representation
theoretical point of view, which in turns leads to new combinatorial properties
of binary trees.Comment: 49 page
Coloured peak algebras and Hopf algebras
For a finite abelian group, we study the properties of general
equivalence relations on G_n=G^n\rtimes \SG_n, the wreath product of with
the symmetric group \SG_n, also known as the -coloured symmetric group. We
show that under certain conditions, some equivalence relations give rise to
subalgebras of \k G_n as well as graded connected Hopf subalgebras of
\bigoplus_{n\ge o} \k G_n. In particular we construct a -coloured peak
subalgebra of the Mantaci-Reutenauer algebra (or -coloured descent algebra).
We show that the direct sum of the -coloured peak algebras is a Hopf
algebra. We also have similar results for a -colouring of the Loday-Ronco
Hopf algebras of planar binary trees. For many of the equivalence relations
under study, we obtain a functor from the category of finite abelian groups to
the category of graded connected Hopf algebras. We end our investigation by
describing a Hopf endomorphism of the -coloured descent Hopf algebra whose
image is the -coloured peak Hopf algebra. We outline a theory of
combinatorial -coloured Hopf algebra for which the -coloured
quasi-symmetric Hopf algebra and the graded dual to the -coloured peak Hopf
algebra are central objects.Comment: 26 pages latex2
Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
The aim of this work is to study the quotient ring R_n of the ring
Q[x_1,...,x_n] over the ideal J_n generated by non-constant homogeneous
quasi-symmetric functions. We prove here that the dimension of R_n is given by
C_n, the n-th Catalan number. This is also the dimension of the space SH_n of
super-covariant polynomials, that is defined as the orthogonal complement of
J_n with respect to a given scalar product. We construct a basis for R_n whose
elements are naturally indexed by Dyck paths. This allows us to understand the
Hilbert series of SH_n in terms of number of Dyck paths with a given number of
factors.Comment: LaTeX, 3 figures, 12 page
- …