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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≤ij≤n and π(i)>π(j). The number of descents is the number of i in the range 1≤in 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 α n times in the multiset for a fixed α with 0α<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.

Normal Approximations for Descents and Inversions of Permutations of Multisets