• Just brute-force the number of configurations for units \(A\) and \(B\) across a few small values of \(c\). Then use polynomial interpolation to get their respective chromatic polynomials.
  • Searching around for “clique sum” and “chromatic polynomial”, bring up a paper detailing how to glue graphs along a shared complete subgraph.
  • Because the specific sequence of \(A\) and \(B\) units in the chain doesn’t change the base number of colouring, the total is just the colourings for a single arrangement multiplied by the number of possible permutations, which is \(\binom{a + b}{a}\). Can also just use DP.