We examine the number of cycles of length in a permutation as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows us to study the 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 , and estimates for the cases of shorter cycles.
"The probability of long cycles in interchange processes." Duke Math. J. 162 (9) 1567 - 1585, 15 June 2013. https://doi.org/10.1215/00127094-2266018