Longest increasing subsequences in involutions avoiding patterns of length three

Abstract

In this note, we study the mean length of the longest increasing subsequence of a uniformly sampled involution that avoids the pattern 3412 and another pattern.

The longest increasing subsequence in involutions avoiding 3412 and another pattern

“…As shown in section 3.2 of [10], E n,T (L n ) = n+1 2 for T = {3412, 231} and T = {3412, 312}. Thus, Theorem 1.1 covers all possible cases for I n (3412, τ ) with τ ∈ S 3 .…”

“…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…”

The feasible regions for consecutive patterns of pattern-avoiding permutations

“…• The most classical one is to look at the limits of various statistics for pattern-avoiding permutations. For instance, the limit distributions of the longest increasing subsequences in uniform pattern-avoiding permutations have been considered in [DHW03,MY19]. Another example is the general problem of studying the limiting distribution of the number of occurrences of a fixed pattern π in a uniform random permutation avoiding a fixed set of patterns when the size tends to infinity (see for instance Janson's papers [Jan17,Jan18a,Jan18b], where the author studied this problem in the model of uniform permutations avoiding a fixed family of patterns of size three).…”