Large Deviations for Permutations Avoiding Monotone Patterns

Abstract

For a given permutation τ , let P τ N be the uniform probability distribution on the set of N -element permutations σ that avoid the pattern τ . For τ = µ k := 123 · · · k, we consider P µ k N (σ I = J) where I ∼ γN and J ∼ δN for γ, δ ∈ (0, 1). If γ + δ = 1 then we are in the large deviations regime with the probability decaying exponentially, and we calculate the limiting value of P µ k N (σ I = J) 1/N . We also observe that for τ = λ k, := 12 . . . k(k − 1) . . . ( + 1) and γ + δ < 1, the limit of P τ N (σ I = J) 1/N is the same as for τ = µ k . arXiv:1606.07906v1 [math.CO]

Longest Monotone Subsequences and Rare Regions of Pattern-Avoiding Permutations

“…For i ∈ [r], let b i be the number of left-to-right maxima of σ in J Ψ i . Observe that if Equation (19) holds, then a n +h i + (i − 1)M + i − 1 is a left-to-right maximum of σ in J Ψ i . Also note that there are at most δn + r left-to-right maxima of σ in (a n , b n + r).…”

“…It is known that r ↑ is the identity function r ↑ (x) = x for the monotone pattern τ = k(k −1) · · · 1, as well as for some other patterns [19]. So far, there is no pattern τ ∈ S k with τ 1 = k for which we can prove that r ↑ is not the identity function.…”

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].…”

Asymptotic distribution of fixed points of pattern-avoiding involutions

“…Recently there has been a growing interest in the statistical properties of random pattern-avoiding permutations. Many of the results concern the overall shape and structure of these permutations [1,12,17,16,19], some explore pattern containment [15], while others consider pattern-avoidance under non uniform distributions such as the Mallows distribution [4]. In [11,13] the limiting distribution on the number and location of fixed points is given for a variety of patternavoiding classes.…”