Distribution of Increasing ℓ-sequences in a Random Permutation

Brad C. Johnson ()
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Brad C. Johnson: University of Manitoba

Methodology and Computing in Applied Probability, 2001, vol. 3, issue 1, 35-49

Abstract: 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.

Keywords: permutations; increasing ℓ-sequences (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1011414107588

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