The authors consider the length, lN , of the length of the longest increasing subsequence of a random permutation of N numbers. The main result in this paper is a proof that the distribution function for lN , suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is ...

We show that a wide variety of generalized increasing subsequence problems admit a one parameter family of extensions for which we can exactly compute the mean length of the longest increasing subsequence. By the nature of the extension, this gives upper bounds on the mean in the unextended model, which turn out to be asymptotically tight for all of the models that have so far been analyzed. A ...

Given a string S = a1a2a3 · · · an, the longest increasing subsequence (LIS) problem is to find a subsequence of S such that the subsequence is increasing and its length is maximal. In this paper, we propose and solve two variants of the LIS problem. The first one is the minimal height LIS where the height means the difference between the greatest and smallest elements. We propose an algorithm ...

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence, that is increasing or decreasing, has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as a polygonal path for which every maximal sub-path, with positiveor negative-slope edges, has at least three points. Given a sequence of distinct r...

This paper is concerned with certain connections between the ensemble of n × n unitary matrices—specifically the characteristic function of the random variable tr(U)—and combinatorics—specifically Ulam’s problem concerning the distribution of the length of the longest increasing subsequence in permutation groups—and the appearance of Painlevé functions in the answers to apparently unrelated que...

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley last-passage percolation model with i.i.d. random weights, and the existence, ergodicity and uniqueness of equilibrium (or ti...

BACKGROUND The potential for reduced pollination ecosystem service due to global declines of bees and other pollinators is cause for considerable concern. Habitat degradation, destruction and fragmentation due to agricultural intensification have historically been the main causes of this pollinator decline. However, despite increasing and accelerating levels of global urbanization, very little ...

oncerns about urbanization and economic developthe question of how the causality works. Economic de_ ment go hand in hand, for a number of reasons. velopment makes it possible for people to produce more. Beyond the question of whether urbanization is a cause Why should this possibility lead to urbanization? Can or a consequence of development, it is the backdrop for our understanding of the lin...

Along the global trend of economic development, it is often observed rapid industrial transformation accompanied by contitual rural-urban migration. In many developing countries there are yet abundant supplies of “surplus labor.”We construct a small open, dynamic framework to examine how the existence of this large supply of rural, unskilled labor a¤ects trade, urbanization, capital accumulatio...