The numbers of ascending segments in a random permutation and in one inverse to it are asymptotically independent

Let $\sigma =\sigma (1)\sigma (2)\ldots\sigma (n)$ be a permutation of the elements of the set $1,2,\ldots,n$, and $D = \{k\colon \sigma ( k ) > \sigma ( k+ 1) \}$ be the descendent set of $\sigma$. Denote by $\des \sigma$ the cardinality of $D$ and set \[ \maj \sigma = \sum_{k\in D} k, \quad \ides \sigma = \des \sigma^{-1}, \quad \imaj \sigma = \maj \sigma^{-1} , \] where $\sigma^{-1}$ is the inverse permutation to $\sigma$. We show that the distribution of the four-dimensional vector $R( n ) = (\des \sigma,\maj \sigma, \ides \sigma, \imaj \sigma)$ is asymptotically normal as $n \to \infty$, and the two first coordinates of $R(n )$ are asymptotically independent from the two last ones.
This work was supported by the Russian Foundation of Basic Research, Grant 93–011–1443.