Exact Largest and Smallest Size of Components |
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Authors: | D Panario B Richmond |
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Affiliation: | (1) School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Canada K1S 5B6. daniel@math.carleton.ca., CA;(2) Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada N2L 3G1. lbrichmond@watdragon.uwaterloo.ca., CA |
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Abstract: | Golomb and Gaal 15] study the number of permutations on n objects with largest cycle length equal to k . They give explicit expressions on ranges n/(i+1) < k ≤ n/i for i=1,2, \ldots, derive a general recurrence for the number of permutations of size n with largest cycle length equal to k , and provide the contribution of the ranges (n/(i+1),n/i] for i=1,2,\ldots, to the expected length of the largest cycle.
We view a cycle of a permutation as a component. We provide exact counts for the number of decomposable combinatorial structures
with largest and smallest components of a given size. These structures include permutations, polynomials over finite fields,
and graphs among many others (in both the labelled and unlabelled cases). The contribution of the ranges (n/(i+1),n/i] for i=1,2,\ldots, to the expected length of the smallest and largest component is also studied.
Received June 27, 2000; revised October 8, 2000. |
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Keywords: | , Largest and smallest components, Random decomposable combinatorial structures, Exponential class, |
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