The paper proves Starr's conjecture on the asymptotics of A(ℓ,n,k), the number of commuting ℓ-tuples of permutations with k joint orbits, and derives log-concavity for typical k.
Abdesselam, Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations
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Proof of a conjecture by Starr and log-concavity for random commuting permutations
The paper proves Starr's conjecture on the asymptotics of A(ℓ,n,k), the number of commuting ℓ-tuples of permutations with k joint orbits, and derives log-concavity for typical k.