A note on the distribution of the sum of lengths of the initial longest increasing sequences in cycles of random permutations

Abstract

Let $S_n$ be the set of all permutations of $\{1,2,\ldots,n\}$ and let $\sigma=(\sigma_1,\sigma_2,\ldots,\sigma_n)\in S_n$. The {\it initial longest increasing sequence} (ILIS) in $\sigma$ has length $m$ if, for $1\le m\le n-1$, $\sigma_1<\sigma_2<\ldots<\sigma_m, \sigma_m>\sigma_{m+1}$, and has length $n$ if $\sigma=(1,2,\ldots,n)$. Let $l(\sigma)$ be the length of the ILIS in $\sigma$. We assume that $\sigma$ is represented in cycle notation, so that the first number in each cycle is the minimum number of this cycle. We also assume that $\sigma$ is chosen uniformly at random from $S_n$, i.e., with probability $1/n!$. Let $C_n(\sigma)$ be the set of all cycles of $\sigma$. 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 $s_n=\sum_{c\in C_n(\sigma)} l(c)$ 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 $n\to\infty$.