A number n is said to be ordinary if the smallest number with exactly n divisors is p q1−1 1 · · · p qa−1 a where q1 · · · qa is the prime factorization of n and q1 ≥ . . . ≥ qa (and where pk denotes the k-th prime). We show here that all square–free numbers are ordinary and that the set of ordinary numbers has natural density one.

P (n, s) denotes the number of permutations of 1, 2,. .. n that have exactly s sequences. Canfield and Wilf [math.CO/0609704] recently showed that P (n, s) can be written as a sum of s polynomials in n. We determine these polynomials explicitly and also obtain explicit expressions for P (n, s) and its fixed-s generating function us(x).