Near-central permutation factorization and Strahovʼs generalized Murnaghan–Nakayama rule

The ( p , q , n ) -dipole problem is a map enumeration problem, arising in perturbative Yang–Mills theory, in which the parameters p and q, at each vertex, specify the number of edges separating of two distinguished edges. Combinatorially, it is notable for being a permutation factorization problem which does not lie in the centre of C [ S n ] , rendering the problem inaccessible through the character-theoretic methods often employed to study such problems. This paper gives a solution to this problem on all orientable surfaces when q = n − 1 , which is a combinatorially significant special case: it is a near-central problem. We give an encoding of the ( p , n − 1 , n ) -dipole problem as a product of standard basis elements in the centralizer Z 1 ( n ) of the group algebra C [ S n ] with respect to the subgroup S n − 1 . The generalized characters arising in the solution to the ( p , n − 1 , n ) -dipole problem are zonal spherical functions of the Gelʼfand pair ( S n × S n − 1 , diag ( S n − 1 ) ) and are evaluated explicitly. This solution is used to prove that, for a given surface, the numbers of ( p , n − 1 , n ) -dipoles and ( n + 1 − p , n − 1 , n ) -dipoles are equal, a fact for which we have no combinatorial explanation. These techniques also give a solution to a near-central analogue of the problem of decomposing a full cycle into two factors of specified cycle type.