Increasing and Decreasing Subsequences of Permutations and Their Variants Richard P. Stanley Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, USA
[email protected] version of 29 November 2005 Abstract.We survey the theory of increasing and decreasing subsequences of permuta- tions. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2,...,n was obtained by Vershik-Kerov and (al- most) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permuta- tions, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions. 1. Introduction Let Sn denote the symmetric group of all permutations of [n] := 1, 2,...,n . We write { } permutations w Sn as words, i.e., w = a1a2 an, where w(i) = ai. An increasing ∈ · · · subsequence of w is a subsequence ai ai satisfying ai < <ai , and similarly for 1 · · · k 1 · · · k decreasing subsequence. For instance, if w = 5642713, then 567 is an increasing subse- quence and 543 is a decreasing subsequence. Let is(w) (respectively, ds(w)) denote the length (number of terms) of the longest increasing (respectively, decreasing) subsequence of w.