Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences

Abstract

We study the longest increasing subsequence problem for random permutations from S n (312, τ ), the set of all permutations of length n avoiding the pattern 312 and another pattern τ , under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutations specifically when the pattern τ is monotone increasing or decreasing, or any pattern of length four.2010 Mathematics Subject Classification. 05A05, 05A15.

Variations on Hammersley’s interacting particle process

“…Three Tracy-Widom distributions (GUE, GOE, GSE) from random matrix ensembles also appear as the limiting distributions for various subsequence problems for permutations [4,8]. For a given pattern τ ∈ S k , the length of the longest τ -avoiding subsequence problem vastly generalizes the longest increasing subsequence problem and leads to many interesting research directions, see [1,11] for a review of the recent results. This paper will motivate further research in this direction and provide insights on the limiting distribution of the proposed models.…”

The longest increasing subsequence in involutions avoiding 3412 and another pattern

“…Variations of Ulam's problem have been considered also for permutations in S n avoiding certain patterns [4,10,11,12]. For permutations π = π 1 π 2 • • • π k ∈ S k and σ = σ 1 σ 2 • • • σ n ∈ S n , we say that σ contains pattern π if there exist 1…”

Longest increasing subsequences in involutions avoiding patterns of length three

“…Variations of Ulam's problem have been considered also for permutations in S n avoiding certain patterns [3,8,9,10]. For permutations π = π 1 π 2 • • • π k ∈ S k and σ = σ 1 σ 2 • • • σ n ∈ S n , we say that σ contains pattern π if there exist 1 ≤ i 1 < i 2 < • • • < i k ≤ n such that σ is < σ it if and only if π s < π t for all 1 ≤ s, t ≤ k.…”