It was recently discovered by Baik, Deift and Johansson [4] that the asymptotic distribution of the length of the longest increasing subsequence in a permutation chosen uniformly at random from Sn, properly centred and normalised, is the same as the asymptotic distribution of the largest eigenvalue of an n × n GUE random matrix, properly centred and normalised, as n → ∞. This distribution had e...

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.

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...

Abstract. We compute the limit distribution for (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 two–colored case our method provides a different proof of a similar result by Tracy and Widom about...

We prove a formula for the number of permutations in Sn such that their first n−k entries are increasing and their longest increasing subsequence has length n − k. This formula first appeared as a consequence of character polynomial calculations in the work of Adriano Garsia, [2]. We give an elementary proof of this result and also of its q-analogue. In [2], Adriano Garsia derived as a conseque...

which consists of an increasing sequence of m decreasing sequences, each of length n/m. The longest increasing subsequence has length m but but the only way for the sample to imply that the input is not sorted is if two chosen elements land in the same decreasing sequence. The probability that this happens is at most ( s 2 ) /m ≤ s/(2m) which is o(1) if s is o( √ m). In particular, for ε = 1/2 ...

Given a distribution G over labeled bipartite (multi) graphs, G = (W; M; E) where jWj = jMj = n, let L(n) denote the size of the largest planar matching of G (here W and M are posets drawn on the plane as two ordered rows of nodes, an upper and a lower one, and a (w; m) edge is drawn as a straight line between w and m). The main focus of this work is to understand the asymptotic (in n) behavior...