Exact Largest and Smallest Size of Components

Algorithmica (Impact Factor: 0.57). 11/2001; 31(3):413-432. DOI: 10.1007/s00453-001-0047-1
Source: DBLP

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.

Key words. Largest and smallest components, Random decomposable combinatorial structures, Exponential class.

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