  We initiate a Stein's method approach to the study of the Plancherel measure
of the symmetric group. A new proof of Kerov's central limit theorem for
character ratios of random representations of the symmetric group on
transpositions is obtained; the proof gives an error term. The construction of
an exchangeable pair needed for applying Stein's method arises from the theory
of harmonic functions on Bratelli diagrams. We also find the spectrum of the
Markov chain on partitions underlying the construction of the exchangeable
pair. This yields an intriguing method for studying the asymptotic
decomposition of tensor powers of some representations of the symmetric group.
