Character-theoretic Techniques for Near-central Enumerative Problems

Abstract

The centre of the symmetric group algebra $\mathbb{C}[\mathfrak{S}_n]$ 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 $\mathfrak{S}_n$-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 \cite{VisentinWieler:2007} associated with Berenstein-Maldacena-Nastase operators in Yang-Mills theory \cite{ConstableFreedmanHeadrick:2002}, for which $k=2.$ Although distinguishing these elements obstructs the use of central methods, these problems may be encoded algebraically in the centralizer of $\mathbb{C}[\mathfrak{S}_n]$ with respect to the subgroup $\mathfrak{S}_{n-k}.$ We develop methods for studying these problems for $k=1,$ and demonstrate their efficacy on the star factorization problem. In a subsequent paper \cite{JacksonSloss:2011}, we consider a special case of the the above dipole problem by means of these techniques.