In a famous paper 8] Hammersley investigated the length L n of the longest increasing subsequence of a random n-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly \soft" arguments that limn ?1=2 EL n = 2. This is a known result, but previous proofs (Vershik-...

We compute the limit distribution for the (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In the two–colored case our method provides a different proof of a similar result by Tracy and Widom about t...

Connections between longest increasing subsequences in random permutations and eigenvalues of random matrices with complex entries have been intensely studied. This note applies properties of random elements of the finite general linear group to obtain results about the longest increasing subsequence in non-uniform random permutations.

Random increasing k-trees represent an interesting, useful class of strongly dependent graphs that have been studied widely, including being used recently as models for complex networks. We study in this paper an informative notion called connectivity-profile and derive, by several analytic means, asymptotic estimates for its expected value, together with the limiting distribution in certain ca...

A straight-line drawing Γ of a graph G = (V,E) is a drawing of G in the Euclidean plane, where every vertex in G is mapped to a distinct point, and every edge in G is mapped to a straight line segment between their endpoints. A path P in Γ is called increasing-chord if for every four points (not necessarily vertices) a, b, c, d on P in this order, the Euclidean distance between b, c is at most ...

We introduce two different kind of increasing bilabellings of trees, for which we provide enumeration formulæ. One of the bilabelled tree families considered is enumerated by the reduced tangent numbers and is in bijection with a tree family introduced by Poupard [11]. Both increasing bilabellings naturally lead to hook-length formulas for trees and forests; in particular, one construction give...

A tree functional is called additive if it satisfies a recursion of the form F (T ) = ∑k j=1 F (Bj)+f(T ), where B1, . . . , Bk are the branches of the tree T and f(T ) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalised planeoriented increasin...

This paper finds and analyzes a formula for the total variation distance between iterations of riffle shuffles and iterations of “cut and then riffle shuffle”. This allows one to obtain information about the convergence rate of permutation statistics (such as the length of the longest increasing subsequence or position of a given card) under these processes. Similar results are given for affine...