Normal Approximations for Descents and Inversions of Permutations of Multisets

Normal approximations for descents and inversions of permutations of the set {1, 2, . . . , n} are well known. We consider the number of inversions of a permutation π(1), π(2), . . . , π(n) of a multiset with n elements, which is the number of pairs (i, j) with 1 ≤ i < j ≤ n and π(i) > π(j). The number of descents is the number of i in the range 1 ≤ i < n such that π(i) > π(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 → ∞, provided that the permutation is equally likely to be any possible permutation of the multiset and no element occurs more than αn times in the multiset for a fixed α with 0 < α < 1. Both normal approximation theorems are proved using the size bias version of Stein’s method of auxiliary randomization and are accompanied by error bounds.