Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 50 min ago
PreprintCC BY 4.0arXiv

On c-polynomial factorisations

  • Harry Hylock
  • Matthew C. Lettington
  • Karl Michael Schmidt
Open at sourcePDFDOICite

Abstract

c-Polynomials are polynomials whose (non-zero) coefficients are all equal to 1. Taking a theorem by Carlitz and Moser as a motivation, we prove a structure theorem for the factorisations of the c-polynomial (xn−1)/(x−1) into c-irreducible factors by relating c-torisations into c-polynomials to joint ordered factorisations arising from the integer n. The question of how many different joint ordered factorisations there are leads to the precise chromatic polynomial for path graphs, counting the ways of path graph colourings with m colours under the condition that all colours are used. Moreover, we give similar counting formulae for joint ordered factorisations under the constraints that all factors are primes, or that all factors are square-free.

Related

  1. Polynomial and Eigenspace Factorisations for the One-loop Dilatation Operator at Finite N
    Adrian Padellaro, Sanjaye Ramgoolam, Ryo Suzuki
    arXivPreprintCC BY 4.0Full textMathematical physicsHigh-energy & particle physicsarXiv
  2. Perfect 1-factorisations of K18 with a non-trivial automorphism group
    Andrei V. Ivanov, Leonid M. Tertitski
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. The Tutte polynomial determines the linear coefficient of the Ehrhart polynomial
    Erik Panzer
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. Weak Polynomial Completeness Does Not Imply Almost Polynomial Completeness
    Viet-Hoang Tran, Tan N. Nguyen
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Explicit M-polynomial and degree-based topological indices of Hanoi graphs
    El-Mehdi Mehiri
    Discrete Applied MathematicsPeer-reviewedLicence not statedFull text2 citedTheoretical & computational chemistryCombinatorics & discrete mathematicsHAL

Cite

BibTeXRIS
BibTeX
@misc{hylock2026cpolynomial,
  title = {{On c-polynomial factorisations}},
  author = {Harry Hylock and Matthew C. Lettington and Karl Michael Schmidt},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.38567},
  eprint = {2609.38567},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38567},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On c-polynomial factorisations
AU  - Harry Hylock
AU  - Matthew C. Lettington
AU  - Karl Michael Schmidt
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.38567
UR  - https://arxiv.org/abs/2609.38567
AB  - c-Polynomials are polynomials whose (non-zero) coefficients are all equal to 1. Taking a theorem by Carlitz and Moser as a motivation, we prove a structure theorem for the factorisations of the c-polynomial (x^n−1)/(x−1) into c-irreducible factors by relating c-torisations into c-polynomials to joint ordered factorisations arising from the integer n. The question of how many different joint ordered factorisations there are leads to the precise chromatic polynomial for path graphs, counting the ways of path graph colourings with m colours under the condition that all colours are used. Moreover, we give similar counting formulae for joint ordered factorisations under the constraints that all factors are primes, or that all factors are square-free.
ER  -