发表机构
Cardiff School of Mathematics（卡迪夫数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文受Carlitz-Moser定理启发，将c-多项式$(x^n-1)/(x-1)$的c-不可约分解与整数n的联合有序分解关联，证明结构定理，并给出路径图染色计数及素数、无平方因数约束下的计数公式。
AI 中文摘要
c-多项式是其（非零）系数均等于1的多项式。以Carlitz和Moser的一个定理为动机，我们通过将c-多项式分解为c-不可约因式的c-分解与由整数n产生的联合有序分解联系起来，证明了c-多项式$(x^n-1)/(x-1)$分解为c-不可约因式的结构定理。关于存在多少种不同的联合有序分解的问题，引出了路径图的精确色多项式，该多项式计数在满足使用所有m种颜色的条件下，用m种颜色对路径图进行染色的方式数。此外，我们在所有因式均为素数或所有因式均为无平方因数的约束下，给出了联合有序分解的类似计数公式。
英文摘要
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.
