On log-concavity of the number of orbits in commuting tuples of permutations

Denote by A(p, n, k) the number of commuting p-tuples of permutations on [n] that have exactly k distinct orbits. It was conjectured in Abdesselam (Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations, http://arxiv.org/abs/2309.07358, 2023) that A(p, n, k) is log-concave with respect to k for every \(p\ge 2, n\ge 3\) $p\ge 2,n\ge 3$ , and the log-concavity was proved in “ \(p=\infty \) $p=\infty$ ” case. In this paper, we prove that for \(k=n-\alpha \) $k=n-\alpha$ , the log-concavity for A(p, n, k) holds for every \(p\ge 2\) $p\ge 2$ for sufficiently large n.