Abstract
AbstractLet $$A(\ell ,n,k)$$
A
(
ℓ
,
n
,
k
)
denote the number of $$\ell $$
ℓ
-tuples of commuting permutations of n elements whose permutation action results in exactly k orbits or connected components. We provide a new proof of an explicit formula for $$A(\ell ,n,k)$$
A
(
ℓ
,
n
,
k
)
which is essentially due to Bryan and Fulman, in their work on orbifold higher equivariant Euler characteristics. Our proof is self-contained, elementary, and relies on the construction of an explicit bijection, in order to perform the $$\ell +1\rightarrow \ell $$
ℓ
+
1
→
ℓ
reduction. We also investigate a conjecture by the first author, regarding the log-concavity of $$A(\ell ,n,k)$$
A
(
ℓ
,
n
,
k
)
with respect to k. The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula.
Publisher
Springer Science and Business Media LLC
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