[Paper Review] On Three Different Notions of Monotone Subsequences

Miklós Bóna|ArXiv.org|Nov 27, 2007

Advanced Combinatorial Mathematics18 references7 citations

TL;DR

This paper compares three distinct notions of monotone subsequences in permutations—standard pattern avoidance, tight containment, and very tight containment—providing enumerative formulas, asymptotic distributions, and proving that the number of very tight 12-patterns in random permutations converges to a Poisson(1) distribution. It establishes that for most patterns of length k, the number of very tight copies is bounded above by that of the monotone pattern αₖ.

ABSTRACT

We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and provide some new results related to limiting distributions.

Motivation & Objective

• To compare three distinct definitions of monotone subsequences in permutations: standard pattern avoidance, tight containment, and very tight containment.
• To analyze the enumerative behavior of permutations avoiding these different notions of monotone patterns, especially for patterns of length 3, 4, and k.
• To investigate the limiting distribution of the number of very tight copies of a pattern, particularly for the 12-pattern, and compare it to the behavior of standard and tight copies.
• To prove that for almost all patterns of length k, the number of very tight copies is bounded above by that of the monotone pattern αₖ = 12⋯k.
• To extend known results on pattern avoidance and distributional limits, particularly in the context of symmetric functions and moment methods.

Proposed method

• Uses the standard definition of pattern avoidance: a permutation p contains pattern q if there exists a subsequence of entries in p with the same relative order as q.
• Applies the Simion-Schmidt bijection to prove that Sₙ(132) = Cₙ, the nth Catalan number, and extends this to show Sₙ(q) = Cₙ for all patterns q of length 3.
• Employs generating functions and symmetric functions, particularly Ira Gessel’s formula for Sₙ(1234), to derive exact enumerative results.
• Applies the method of moments to prove convergence in distribution: shows that the j-th moment of the number of very tight 12-patterns converges to the j-th moment of a Poisson(1) distribution.
• Uses indicator random variables Zₙ,ᵢ for the event that pᵢ + 1 = pᵢ₊₁, and computes E(Zₙʲ) to analyze the limiting distribution.
• Proves an upper bound on Vₙ(q), the number of very tight copies of pattern q, by comparing it to P(n,i,αₖ), the probability of very tight copies of the monotone pattern αₖ.

Experimental results

Research questions

• RQ1 How do the three notions of monotone subsequences—standard, tight, and very tight—differ in terms of enumeration and distributional behavior?
• RQ2 Why does Sₙ(1342) differ from Sₙ(1234) and Sₙ(1324) for n ≥ 6, despite all being patterns of length 4?
• RQ3 What is the limiting distribution of the number of very tight copies of a fixed pattern, particularly for the 12-pattern?
• RQ4 For which patterns q of length k is Vₙ(q) ≤ Vₙ(αₖ), and how common are such patterns as k increases?
• RQ5 Can the method of moments be used to prove convergence to a Poisson distribution for the number of very tight 12-patterns in random permutations?

Key findings

• For all patterns q of length 3, the number of n-permutations avoiding q is the nth Catalan number Cₙ, regardless of the specific pattern.
• For patterns of length 4, Sₙ(1342), Sₙ(1234), and Sₙ(1324) are distinct sequences, with Sₙ(1342) being significantly smaller than Sₙ(1234) for n ≥ 6.
• An exact formula for Sₙ(1234) is given using symmetric functions: Sₙ(1234) = (1/(n+1)²(n+2)) × Σₖ₌₀ⁿ (2k choose k)(n+1 choose k+1)(n+2 choose k+1).
• The formula for Sₙ(1342) is surprising: it involves alternating sums and factorials, and is not a simple rational function or hypergeometric series.
• The number of very tight copies of the 12-pattern in a random n-permutation converges in distribution to a Poisson(1) distribution as n → ∞.
• For almost all patterns of length k (in the limit as k → ∞), the number of very tight copies is bounded above by that of the monotone pattern αₖ = 12⋯k.