Chapter 33 Longest Increasing Subsequence
نویسنده
چکیده
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 construction and random matrices. The presentation in this paper is largely modeled on that of Aldous and Diaconis [2], but is shorter and designed to be read by a wider audience. A reader with two semesters of abstract algebra and some probability should be able to follow the ideas presented here with little difficulty. The problem in which we are interested in is as follows. Let π be a random permutation of the integers 1, 2, . . . , n. π(i) is the i-th element of the permutation. Then an increasing subsequence (i1, i2, . . . , ik) of π is a subsequence satisfying:
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