A note on the distribution of the sum of lengths of the initial longest increasing sequences in cycles of random permutations
Ljuben Mutafchiev
TL;DR
The paper analyzes the distribution of the sum of the ILIS lengths across cycles of a random permutation and proves a central limit theorem for the normalized statistic $s_n' = \frac{s_n - e\log n}{\sqrt{3e\log n}}$. It builds on Mansour's exponential generating function for $\mathbb{E}(y^{s_n})$, and combines Darboux-type singularity analysis with Curtiss' continuity theorem to obtain a normal limit. The result provides a probabilistic refinement of Mansour's exact and asymptotic findings for $s_n$ and illuminates the interplay between cycle structure and ILIS statistics in random permutations. This clarifies the fluctuation behavior of cycle-based permutation statistics in the asymptotic regime and demonstrates the use of analytic combinatorics techniques in deriving distributional limits.
Abstract
Let $S_n$ be the set of all permutations of $\{1,2,\ldots,n\}$ and let $σ=(σ_1,σ_2,\ldots,σ_n)\in S_n$. The {\it initial longest increasing sequence} (ILIS) in $σ$ has length $m$ if, for $1\le m\le n-1$, $σ_1<σ_2<\ldots<σ_m, σ_m>σ_{m+1}$, and has length $n$ if $σ=(1,2,\ldots,n)$. Let $l(σ)$ be the length of the ILIS in $σ$. We assume that $σ$ is represented in cycle notation, so that the first number in each cycle is the minimum number of this cycle. We also assume that $σ$ is chosen uniformly at random from $S_n$, i.e., with probability $1/n!$. Let $C_n(σ)$ be the set of all cycles of $σ$. In [9], T. Mansour investigated enumerative properties related to lengths of the ILIS in random permutations represented by the cycle notation. In particular, he studied the sum of the ILIS' lengths defined by $s_n=\sum_{c\in C_n(σ)} l(c)$ and derived exact and asymptotic expressions for its expectation and variance. In this note, we supplement Mansour's results on $s_n$ with a limit theorem. We show that $s_n$, appropriately normalized, converges weakly to a standard normal random variable as $n\to\infty$.