Evaluations of the Circuit Partition Polynomial

J. A. Ellis-Monaghan (1998, J. Combin. Theory Ser. B74, 326–352) used Hopf algebra techniques to prove some beautiful combinatorial interpretations of the Martin polynomial for unoriented graphs. Our aim here is to give very simple proofs of similar interpretations for a considerably wider class of values. The results look particularly simple when formulated for a trivial transform of the Martin polynomial, the circuit partition polynomial.