A Generalization of Redfield's Master Theorem

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Generalizations 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 Sd into transitive constituents. As direct consequences, one obtains several graphical corollaries. Given graphs Γ₁,,Γₖ, with d vertices, together with their automorphism groups W₁≤ Sd,, Wₖ≤ Sd, one can find the number of superpositions of Γ₁,,Γₖ, 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 Wₘ's is cyclic; (3) the groups are of odd order, in case at least one of Wₘ'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 Wₘ'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 Wₘ's has a normal q-subgroup such that the corresponding factor-group is cyclic of order relatively prime to q.