Combinatorial Addition Formulas and Applications

We derive addition formulas at the combinatorial level, that is equations of the form F (X1+ X2 + +Xk) = F (X1; X2; : : : ; Xk), where F = F (X) is a given combinatorial species and F is a species on k sorts of singletons X1; X2; : : : ; Xk; depending on F . General results are given in the case of a molecular species M = X=H. Speci c formulas are also presented in the cases of the species Ln of n-lists, Chan of n-chains, En of n-sets, E n of oriented n-sets, Cn of (oriented) n-cycles and Pn of n-gons (unoriented cyles). These formulas are useful for the computation of molecular expansions of species de ned by functional equations. Applications to the computation of cycle index series and asymmetry index series, to the extension of substitution to virtual species (and K-species), and to the analysis of generalized binomial coe cients M N k for molecular species are also given.