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The Variance and the Asymptotic Distribution of the Length of Longest $k$-alternating Subsequences

Altar Çiçeksiz, Yunus Emre Demirci, Ümit Işlak

TL;DR

An explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation is obtained and an asymptotic formula is obtained for the length of longest longest subsequence in random permutations.

Abstract

We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random permutations. Also a central limit is proved for the latter statistic.

The Variance and the Asymptotic Distribution of the Length of Longest $k$-alternating Subsequences

TL;DR

An explicit formula for the variance of the number of -peaks in a uniformly random permutation is obtained and an asymptotic formula is obtained for the length of longest longest subsequence in random permutations.

Abstract

We obtain an explicit formula for the variance of the number of -peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest -alternating subsequence in random permutations. Also a central limit is proved for the latter statistic.

Paper Structure

This paper contains 3 sections, 5 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. The variances of $P$ and $\operatorname{as}_{n, k}$
  3. A Central Limit theorem

Key Result

Proposition 1.1

Let $\sigma=\sigma(1) \sigma(2) \ldots \sigma(n) \in S_n$, $i \in [n]$ and $1 \leq k \leq n-1$. Then $\sigma(i)$ is a $k$-peak if and only if it satisfies both of the following two properties: (i) If there is an $s>i$ with $\sigma(s)>\sigma(i)$, then there is a $k$-down $\sigma(i)\ldots \sigma(j)$ i

Theorems & Definitions (8)

  • Proposition 1.1
  • Theorem 2.1
  • Corollary 2.1
  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Remark 3.2