  In this paper, we study the relationship between polynomial integrals on the
unitary group and the conjugacy class expansion of symmetric functions in
Jucys-Murphy elements. Our main result is an explicit formula for the top
coefficients in the class expansion of monomial symmetric functions in
Jucys-Murphy elements, from which we recover the first order asymptotics of
polynomial integrals over $\U(N)$ as $N \rightarrow \infty$. Our results on
class expansion include an analogue of Macdonald's result for the top
connection coefficients of the class algebra, a generalization of Stanley and
Olshanski's result on the polynomiality of content statistics on
Plancherel-random partitions, and an exact formula for the multiplicity of the
class of full cycles in the expansion of a complete symmetric function in
Jucys-Murphy elements. The latter leads to a new combinatorial interpretation
of the Carlitz-Riordan central factorial numbers.
