 
Summary: BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 36, Number 4, Pages 413432
S 02730979(99)00796X
Article electronically published on July 21, 1999
LONGEST INCREASING SUBSEQUENCES: FROM PATIENCE
SORTING TO THE BAIKDEIFTJOHANSSON THEOREM
DAVID ALDOUS AND PERSI DIACONIS
Abstract. We describe a simple oneperson card game, patience sorting. Its
analysis leads to a broad circle of ideas linking Young tableaux with the longest
increasing subsequence of a random permutation via the Schensted correspon
dence. A recent highlight of this area is the work of BaikDeiftJohansson
which yields limiting probability laws via hard analysis of Toeplitz determi
nants.
1. Introduction
This survey paper treats two themes in parallel. One theme is a purely math
ematical question: describe the asymptotic law (probability distribution) of the
length of the longest increasing subsequence of a random permutation. This ques
tion has been studied by a variety of increasingly technically sophisticated methods
over the last 30 years. We outline three, apparently quite unrelated, methods in
