Séminaire Lotharingien de Combinatoire, 84B.79 (2020), 12 pp.
Liam Hanany and Doron Puder
Word Measures on Symmetric Groups
Abstract.
Fix a word w in a free group Fr on r generators. A w-random
permutation in the symmetric group Sn is obtained by sampling r independent uniformly
random permutations σ1,...,σr in Sn and evaluating
w(σ1,...,σr). In (Puder 2014, Puder-Parzanchevski 2015)
it was shown that the average number of fixed points in a w-random
permutation is 1+θ(n1-π(w)), where
π(w) is the smallest rank of a subgroup H <= Fr
containing w as a non-primitive element. We show that π(w) plays a role in estimates
of other natural families of characters. In particular, we show that
for all s>=2, the average number of s-cycles is (1/s)+O(n-π(w)).
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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