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...

We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old f...

In this paper, we present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data streaming model. For the problem of deciding whether the LIS of a given stream of integers drawn from {1, . . . ,m} has length at least k, we discuss a one-pass streaming algorithm using O(k log m) space, with update time either O(log k) or...

The longest increasing subsequence of a random walk with mean zero and finite variance is known to be n1/2+o(1). We show that this is not universal for symmetric random walks. In particular, the symmetric fat-tailed random walk has a longest increasing subsequence that is asymptotically at least n0.690 and at most n0.815. An exponent strictly greater than 1/2 is also shown for the symmetric sta...