Enumeration of orbits on cycles for linear and affine groups (Q1399236)

If \(G\) is a permutation group acting on \(n\) elements, the Parker vector \(p(G)=(p_1(G),p_2(G),\dots,\break p_n(G))\) is given by the numbers \(p_k(G)\) of orbits of \(G\) (by conjugation in \(\text{Sym}(k)\)) on the set of cycles of length \(k\) appearing in the cycle decomposition of elements of \(G\). In the present paper, the Parker vectors of the groups \(\text{GL}(m,q)\), \(\text{SL}(m,q)\) and \(\text{AGL}(m,q)\) in their natural actions are determined. In fact, the Parker vectors are closely related to the enumeration of rational normal forms of elements of general linear groups.