Salihu Lawan Aliyu1, Amina Muhammad Lawan2 and Shuaibu Garba Ngulde1
1Department of Mathematical Sciences, University of Maiduguri, P. M. B. 1069, Maiduguri, Borno State, Nigeria
2Department of Mathematical Sciences, Bayero University Kano, Nigeria
*Corresponding author’s Email: salihualiyulawan@gmail.com, doi.org/10.55639/607.02010089
ABSTRACT
We study full r-colourings of a finite chain by interpreting colourings as functions from into a finite colour set. This functional framework enables a systematic analysis of the structure and enumeration of colourings using tools from partition theory, Stirling numbers, and permutation group actions. We decompose the space of all colourings according to image size and show that the number of colourings using exactly is governed by the Stirling numbers of the second kind, yielding a natural Stirling decomposition of the classical identity rn. We further investigate recolourings induced by the natural action of the symmetric group Sn on the domain. By analysing group structures and applying Burnsides Lemma, we derive closed form expressions for both the average and total number of recolourings, which are shown to be given by simple binomial coefficients. These results reveal an unexpected regularity in the global behaviour of recolouring orbits. The action of the alternating group is also examined. Although fixed-point conditions depend only on cycle structure, the restriction to even permutations introduce subtle combinatorial differences. Computational evidence suggests that several enumerative patterns persist in this setting, leading to conjectural formulas for recolourings under An . The results establish a unified link between colouring problems, partition theory, and group actions, and they provide a flexible framework for further extensions to other colouring models and transformation semigroups.
Keywords:
Full colourings,
Group actions,
Stirling numbers,
Burnside’s lemma,
Symmetric group
.

