Jack Derangements Article Swipe

PDF

Related Concepts

Combinatorics Mathematics Simple (philosophy) Lambda Eigenvalues and eigenvectors Discrete mathematics Combinatorial proof Integer sequence Generating function Physics Quantum mechanics Philosophy Epistemology Optics

Nathan Lindzey ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2304.06629 · OA: W4365601430

For each integer partition $λ\vdash n$ we give a simple combinatorial expression for the sum of the Jack character $θ^λ_α$ 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 the principal lower hook product.