Orbit decomposition of subset actions (Q1598806)

Let \(G\) be a permutation group acting on a finite set \(A\). Then \(G\) acts naturally on \(\lambda^r(A)\), the set of \(r\)-element subsets of \(A\), for each positive integer \(r\). In the present paper the orbit decomposition of \(\lambda^r(A)\) is determined by a generating function formula giving the multiplicity of orbits of type \(\{Lx\mid x\in G\}\) for subgroups \(L\) of \(G\) (Theorem 2.5). The formula is obtained by Möbius inversion over the lattice of subgroups.