Character-theoretic techniques for near-central enumerative problems

The centre of the symmetric group algebra C [ S n ] has been used successfully for studying important problems in enumerative combinatorics. These include maps in orientable surfaces and ramified covers of the sphere by curves of genus g, for example. However, the combinatorics of some equally important S n -factorization problems forces k elements in { 1 , … , n } to be distinguished. Examples of such problems include the star factorization problem, for which k = 1 , and the enumeration of 2-cell embeddings of dipoles with two distinguished edges associated with Berenstein–Maldacena–Nastase operators in Yang–Mills theory, for which k = 2 . Although distinguishing these elements obstructs the use of central methods, these problems may be encoded algebraically in the centralizer of C [ S n ] with respect to the subgroup S n − k . We develop methods for studying these problems for k = 1 , and demonstrate their efficacy on the star factorization problem. In a subsequent paper, we consider a special case of the above dipole problem by means of these techniques.