On c-polynomial factorisations

Source 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 $(x^n-1)/(x-1)$ 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.