  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 $\mathbb{C}[\mathfrak{S}_n]$, 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 \emph{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 $\mathbb{C}[\mathfrak{S}_n]$ with
respect to the subgroup $\mathfrak{S}_{n-1}$. The generalized characters
arising in the solution to the $(p,n-1,n)$-dipole problem are zonal spherical
functions of the Gel'fand pair $(\mathfrak{S}_n\times \mathfrak{S}_{n-1},
\mathrm{diag}(\mathfrak{S}_{n-1}))$ 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.
