The Enumeration of Covers of a Finite Set

Symmetric functions and P-recursiveness

“…and Jackson [8], The fundamental fact relating these two concepts is that a(n) is P-recursive if and only if its generating function ∞ n=0 a(n)x n is D-finite. We refer the reader to Stanley [47] for the proof of this and other basic facts.…”

Some connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisets

“…was proved by Bender [2]. The case of general m was proved by Devitt and Jackson [4]. They also prove that there exists a number c such that the extraction of the coefficient v k;m from the exponential generating function V m (x) can be done in no more than ck m log k arithmetic operations.…”