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Normal Approximations for Descents and Inversions of Permutations of Multisets

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Abstract

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 α n 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

    Mark Conger & D. Viswanath

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  1. Mark Conger
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  2. D. Viswanath
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Correspondence to D. Viswanath.

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This work was supported by a research fellowship from the Sloan Foundation.

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Conger, M., Viswanath, D. Normal Approximations for Descents and Inversions of Permutations of Multisets. J Theor Probab 20, 309–325 (2007). https://doi.org/10.1007/s10959-007-0070-5

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  • DOI: https://doi.org/10.1007/s10959-007-0070-5

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