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Distribution of Increasing ℓ-sequences in a Random Permutation

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  • Brad C. Johnson

    (University of Manitoba)

Abstract

This paper examines the distribution of the number, k, of increasing ℓ-sequences in a random permutation of $$\left\{ {1,...,n} \right\}$$ . A new solution is determined based on the compositions of n which requires, at most, $$k\left( {n - k - \ell } \right)$$ summands. This solution easily yields existing results for the special case $$\ell = 2$$ and provides an alternate form for the case $$\ell = 3$$ . The expected number of increasing ℓ-sequences in a random permutation is determined and it is shown that the limiting distribution is degenerate about 0 for $$\ell > 2$$ . An alternate algorithm to determine the exact distribution is presented, based on the partitions of n, which is easy to implement and efficient for small n. Applications in non-parametric statistics and graph theory are discussed.

Suggested Citation

  • Brad C. Johnson, 2001. "Distribution of Increasing ℓ-sequences in a Random Permutation," Methodology and Computing in Applied Probability, Springer, vol. 3(1), pages 35-49, March.
    • Handle: RePEc:spr:metcap:v:3:y:2001:i:1:d:10.1023_a:1011414107588
    DOI: 10.1023/A:1011414107588
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    References listed on IDEAS

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    1. James Fu, 1995. "Exact and limiting distributions of the number of successions in a random permutation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(3), pages 435-446, September.
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    More about this item

    Keywords

    permutations; increasing ℓ-sequences;

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