2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78107Generalizations of Redfield's master theorem and superposition theorem are proved by using decomposition of the tensor product of several induced monomial representations of the symmetric group $S_d$ into transitive constituents. As direct consequences, one obtains several graphical corollaries. Given graphs $Γ_1,\hdots ,Γ_k$, with $d$ vertices, together with their automorphism groups $W_1\leq S_d,\hdots, W_k\leq S_d$, one can find the number of superpositions of $Γ_1,\hdots ,Γ_k$, whose automorphism groups satisfy one of the following conditions: (1) the groups consist of even permutations; (2) the groups are trivial, in case at least one of $W_m$'s is cyclic; (3) the groups are of odd order, in case at least one of $W_m$'s is dihedral and its order is not divisible by 4; (4) the groups are of order dividing a natural number $r$, in case at least one of $W_m$'s has a normal solvable subgroup of order $r$, such that the corresponding factor-group is cyclic of order relatively prime to $r$; (5) the groups are $q$-groups ($q$ is a prime), in case at least one of $W_m$'s has a normal $q$-subgroup such that the corresponding factor-group is cyclic of order relatively prime to $q$.9 pages, uses plain TeXRepresentation TheoryCombinatorics05A15, 05C30text