Longest Monotone Subsequences and Rare Regions of Pattern-Avoiding Permutations

Abstract

We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size n that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or more. In our results, the longest monotone subsequences have expected length proportional to n for pattern-avoiding classes, in contrast with the √ n behaviour that holds for unrestricted permutations.In addition, for a pattern τ of length k, we scale the plot of a random τ -avoiding permutation down to the unit square and study the "rare region," which is the part of the square that is exponentially unlikely to contain any points. We prove that when τ 1 > τ k , the complement of the rare region is a closed set that contains the main diagonal of the unit square. For the case τ 1 = k, we also show that the lower boundary of the part of the rare region above the main diagonal is a curve that is Lipschitz continuous and strictly increasing on [0, 1].

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

“…Note that s n = [x n ]F τ (x, 1). By Corollary 2.4 and Proposition 2.6, we recover some of the relevant results in [24].…”

“…For example, Theorem 3.4 for m = 3 gives that E 321 (L n ) ∼ 3n 4 as shown in [24], and for m = 4, we have…”

“…The longest increasing subsequence problem for the pattern-avoiding permutations was first studied for the patterns of length tree, that is, for τ ∈ S 3 on S n (τ ) with uniform probability distribution in [13]. The case S n (τ 1 , τ 2 ) with τ 1 , τ 2 ∈ S 3 is studied for all possible cases in [24]. One of the corollaries of our main result, Theorem 2.1, covers the case S n (312, τ ) with either τ ∈ S 3 (312) or τ ∈ S 4 (312) and hence add some new results to this research program.…”

“…The paper is organized as follows. We present our main result, Theorem 2.1, in Section 2 and as a first case apply it to S n (312, τ ) with τ ∈ S 3 (312) which gives an alternative proof for some cases considered in [24] through generating functions. In Section 3, we consider three specific longer patterns where τ is monotone increasing/decreasing pattern or the pattern (m − 1)m(m − 2)(m − 3) • • • 321.…”

Longest increasing subsequences in involutions avoiding patterns of length three

“…The problem of determining the asymptotic behavior and limiting distribution of L n on S n under the uniform probability distribution has led to very interesting and important research in the last fifty years, which made some unexpected connections among different fields of mathematics and physics; see [1-3, 7, 8, 13, 18, 26] and references therein. Probabilistic study of pattern-avoiding permutation classes has recently become an active area of research; for some recent works in this direction, see [9,19,[22][23][24].…”