We prove a large deviations principle for the empirical law of the block sizes of a uniformly distributed non-crossing partition. Using well-known bijections we relate this to other combinatorial objects, including Dyck paths, permutations and parking functions. 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.
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