Exact Largest and Smallest Size of Components 1

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, . . . , 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, . . . , 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, . . . , to the expected length of the smallest and largest component is also studied.