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Normal Approximations for Descents and Inversions of Permutations of Multisets

Author

Listed:
  • Mark Conger

    (University of Michigan)

  • D. Viswanath

    (University of Michigan)

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 π(j). The number of descents is the number of i in the range 1≤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

Suggested Citation

  • Mark Conger & D. Viswanath, 2007. "Normal Approximations for Descents and Inversions of Permutations of Multisets," Journal of Theoretical Probability, Springer, vol. 20(2), pages 309-325, June.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:2:d:10.1007_s10959-007-0070-5
    DOI: 10.1007/s10959-007-0070-5
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    References listed on IDEAS

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    1. Byron J. T. Morgan, 1989. "Introduction to Optimization," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 152(2), pages 254-255, March.
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