We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles.

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Abstract. Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order ≺ on the irreducible representations of Sn. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the "Aldous order" completely is a generalized question. We show a few additional entries for this order.

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