Séminaire Lotharingien de Combinatoire, 91B.97 (2024), 12 pp.
Nathan Lindzey
Jack Derangements
Abstract.
For each integer partition λ of n we give a simple
combinatorial expression for the sum of the Jack character
θλα over the integer
partitions of n with no
singleton parts.
For α = 1,2 this gives closed forms for the eigenvalues of the
permutation and perfect matching derangement graphs, resolving an open
question in algebraic graph theory.
A byproduct of the latter is a simple combinatorial formula for the
immanants of the matrix J-I where J is the all-ones matrix, which
might be of independent interest.
Our proofs center around a Jack analogue of a hook product related to
Cayley's Ω-process in classical invariant theory, which we call
\emph{the principal lower hook product}.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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