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...

The effects of black rice powder on the quality of sausage were investigated. Samples were prepared with 0% (control), 1%, 3%, and 5% black rice powder. With increasing black rice powder content, the moisture and ash content of sausage increased, while protein content was significantly less than that observed for the control (p<0.05). The fat content in samples containing 5% black rice powder w...

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...