Normal approximations for descents and inversions of permutations of multisets

Details Normal approximations for descents and inversions of permutations of multisets Normal approximations for descents and inversions of permutations of multisets

2005-08-14

arxiv.org/abs/math/0508242v2

Mathematics

Probability

Scientific paper

Normal approximations for descents and inversions of permutations of the set $\{1,2,...,n\}$ are well known. A number of sequences that occur in practice, such as the human genome and other genomes, contain many repeated elements. Motivated by such examples, we consider the number of inversions of a permutation $\pi(1), \pi(2),...,\pi(n)$ of a multiset with $n$ elements, which is the number of pairs $(i,j)$ with $1\leq i < j \leq n$ and $\pi(i)>\pi(j)$. The number of descents is the number of $i$ in the range $1\leq i < n$ such that $\pi(i) > \pi(i+1)$. We prove that, appropriately normalized, the distribution of both inversions and descents of a random permutation of the multiset approaches the normal distribution as $n\to\infty$, provided that the permutation is equally likely to be any possible permutation of the multiset and no element occurs more than $\alpha n$ times in the multiset for a fixed $\alpha$ with $0<\alpha < 1$. Both normal approximation theorems are proved using the size biased version of Stein's method of auxiliary randomization and are accompanied by error bounds.

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