  We prove a large deviations principle for the empirical law of the block
sizes of a uniformly distributed non-crossing partition. As an application we
obtain a variational formula for the maximum of the support of a compactly
supported probability measure in terms of its free cumulants, provided these
are all non-negative. This is useful in free probability theory, where
sometimes the R-transform is known but cannot be inverted explicitly to yield
the density.
