Random Permutations, Non-Decreasing Subsequences and Statistical Independence

Permutations Without Long Decreasing Subsequences and Random Matrices

ABSTRACT. We study the shape of the Young diagram λ associated via the Robinson–Schensted– Knuth algorithm to a random permutation in Sn such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n → ∞ the rows of λ behave like ...

Partitioning Permutations into Increasing and Decreasing Subsequences

Descending subsequences of random permutations

Given a random permutation of the numbers 1, 2, . . . . n, let L, be the length of the longest descending subsequence of this permutation. Let F, be the minimal header (first element) of the descending subsequences having maximal length. It is known that EL,/&,,,, c and that c=2. However, the proofs that r=2 are far from elementary and involve limit processes. Several relationships between thes...

Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

In a recent paper, Backelin, West and Xin describe a map φ∗ that recursively replaces all occurrences of the pattern k . . . 21 in a permutation σ by occurrences of the pattern (k − 1) . . . 21k. The resulting permutation φ∗(σ ) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map φ∗ commutes with taking the inverse of a permutation. In the BWX paper, the ...

On longest increasing subsequences in random permutations

The expected value of L n , the length of the longest increasing subsequence of a random permutation of f1; : : : ; ng, has been studied extensively. This paper presents the results of both Monte Carlo and exact computations that explore the ner structure of the distribution of L n. The results suggested that several of the conjectures that had been made about L n were incorrect, and led to new...