Complementing the results claiming that the maximal length L n of an increasing subsequence in a random permutation of f1; 2; : : : ; ng is highly concentrated, we show that L n is not concentrated in a short interval: sup l P(l L n l + n 1=16 log ?3=8 n) ! 0 as n ! 1.

Let S = s1, s2, . . . , sn be an integer sequence. The longest increasing subsequence problem is to find an increasing subsequence of S with the longest length. By regarding S as a weight sequence of the vertices in a path, we can redefine the longest increasing subsequence problem on graphs as follows. Let G = (V,E) be a graph in which every vertex v ∈ V has a weight w(v). A longest increasing...

where each ( ) ∈ , 1 ≤  ≤ , and  is arbitrary. For convenience, define (0 0) = (0 0) and (+1 +1) = (1 1). Define such a point sequence to be an up/right path if, for any  ≥ 1, we have −1 ≤  and −1 ≤ . Hence an up/right path joins points of  in a continuous, piecewise linear manner with line segments of slope , 0 ≤  ≤ ∞, attaching (−1 −1) and ( ) for al...

We consider the distribution of the length of the longest subsequence avoiding an arbitrary pattern, π, in a random permutation of length n. The well-studied case of a longest increasing subsequence corresponds to π = 21. We show that there is some constant cπ such that as n → ∞ the mean value of this length is asymptotic to 2 √ cπn and that the distribution of the length is tightly concentrate...

GRACIELA BERBERIÁN,1 CECILIA HIDALGO,2 REINALDO DIPOLO,3 AND LUIS BEAUGÉ1 1Instituto de Investigación Médica Mercedes y Martı́n Ferreyra, 5000 Córdoba, Argentina; 2Centro de Estudios Cientı́ficos de Santiago, Santiago 9; and Departamento de Fisiologı́a y Biofı́sica, Facultad de Medicina, Universidad de Chile, Santiago 7, Chile; and 3Laboratorio de Permeabilidad Iónica, Instituto Venezolano de Inves...

In this paper we will investigate the connection between random matrices and finding the longest increasing subsequence of a permutation. We will introduce a model for the problem using a simple card game. Then we will talk about Young tableaux and their relation to the symmetric group. Representation theory and power-sum symmetric functions serve as the bridge between this combinatorial constr...

A two-rowed array αn = ( a1 a2 . . . an b1 b2 . . . bn ) is said to be in lexicographic order if ak ≤ ak+1 and bk ≤ bk+1 if ak = ak+1. A length ` (strictly) increasing subsequence of αn is a set of indices i1 < i2 < . . . < i` such that bi1 < bi2 < . . . < bi` . We are interested in the statistics of the length of the longest increasing subsequence of αn chosen according to Dn, for distinct fam...

We propose a new nonparametric test for the supposition of independence between two continuous random variables X and Y. Given a sample of (X,Y ), the test is based on the size of the longest increasing subsequence of the permutation which maps the ranks of the X observations to the ranks of the Y observations. We identify the independence assumption between the two continuous variables with th...

We investigate combinatorial enumeration problems related to subsequences of strings; in contrast to substrings, subsequences need not be contiguous. For a finite alphabet Σ, the following three problems are solved. (1) Number of distinct subsequences: Given a sequence s ∈ Σ and a nonnegative integer k ≤ n, how many distinct subsequences of length k does s contain? A previous result by Chase st...

Inspired by the results of Stanley and Widom concerning the limiting distribution of the lengths of longest alternating subsequences in random permutations, and results of Deutsch, Hildebrand and Wilf on the limiting distribution of the longest increasing subsequence for pattern-restricted permutations, we find the limiting distribution of the longest alternating subsequence for pattern-restric...