Exact Largest and Smallest Size of Components

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.