Metadata
bibkey: cameronsemeraro2018cycle
authors: Peter J. Cameron, Jason Semeraro
year: 2018
title: The cycle polynomial of a permutation group
doi: 10.37236/7299
url: https://arxiv.org/abs/1701.06954
claim: "The cycle polynomial sums x to the number of permutation cycles, including fixed points; its reciprocity with orbital chromatic polynomials supplies the group-graph reciprocal-pair problem."
strata_touched:
- D5/S3/Combinatorics/ReciprocalPairs/GroupGraphReciprocalRefutation
license: citation-only
triage: anchor
Verified locator
DOI: 10.37236/7299
Source: https://arxiv.org/abs/1701.06954
Published in Electronic Journal of Combinatorics 25(1) (2018). Campbell cites this paper as reference [2]. Section 1 defines the cycle polynomial; Section 3 relates it to orbital chromatic polynomials. Propositions 3.5 and 3.6 give the product and wreath-product constructions cited in Campbell’s irreducibility discussion.
Mathematical scope
The cycle polynomial is F_G(x) = Σ_{g∈G} x^{c(g)}, where fixed points count as cycles. The reciprocal-pair identity compares F_G(−x), multiplied by (−1)^n, with the sum of the chromatic polynomials of the graphs obtained by contracting permutation cycles. An edge within one contracted cycle produces a loop and contributes zero proper colourings. This is background for the Campbell refutation; the sign-coupled counterexample is a separate result.