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 $P(M,L)$ of the number $M$ 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 $P(M,L)$ 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 $L \to \infty$.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 $n$, 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 $X$ at time $n$, 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 $1,2,3, >...,L$ on sites of a 1d or 2d square lattices containing $L$ 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 $G$ a finite abelian group, we study the properties of general equivalence relations on G_n=G^n\rtimes \SG_n, the wreath product of $G$ with the symmetric group \SG_n, also known as the $G$-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 $G$-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or $G$-coloured descent algebra). We show that the direct sum of the $G$-coloured peak algebras is a Hopf algebra. We also have similar results for a $G$-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 $G$-coloured descent Hopf algebra whose image is the $G$-coloured peak Hopf algebra. We outline a theory of combinatorial $G$-coloured Hopf algebra for which the $G$-coloured quasi-symmetric Hopf algebra and the graded dual to the $G$-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