Longest increasing subsequences and log concavity

2015/11/27 by Miklós Bóna, Bóna, Miklós, Marie-Louise Lackner +3
Mathematics · #05A05 (Primary) #05A20 #05E99 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A05 #msc:05A20 #msc:05E99

paper · pdf · doi:10.48550/arxiv.1511.08653

15 pages, 2 figures

arxiv created 2015/11/27 · arxiv updated 2015/11/30

Abstract

Let π be a permutation of [n]=\1,…,n\ and denote by ℓ(π) the length of a longest increasing subsequence of π. Let ℓ_n,k be the number of permutations π of [n] with ℓ(π)=k. Chen conjectured that the sequence ℓ_n,1,ℓ_n,2,…,ℓ_n,n is log concave for every fixed positive integer n. We conjecture that the same is true if one is restricted to considering involutions and we show that these two conjectures are closely related. We also prove various analogues of these conjectures concerning permutations whose output tableaux under the Robinson-Schensted algorithm have certain shapes. In addition, we present a proof of Deift that part of the limiting distribution is log concave. Various other conjectures are discussed.

Related

• Increasing Subsequences and Kronecker Coefficients
• A combinatorial proof of the log-concavity of the numbers of permutations with k runs
• Log-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations
• Log-concavity, the Ulam distance and involutions
• Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations