An extension of the exponential formula in enumerative combinatorics

Let α be a formal variable and Fw be a weighted species of structures (class of structures closed under weight-preserving isomorphisms) of the form Fw = E(F c w), where E and F c w respectively denote the species of sets and of connected Fw-structures. Multiplying by α the weight of each F c wstructure yields the species Fw(α) = E(F c αw). We introduce a “universal” virtual weighted species, Λ, such that Fw(α) = Λ ◦ F w , where F w denotes the species of non-empty Fw-structures. Using general properties of Λ , we compute the various enumerative power series G(x), G̃(x), G(x), G(x; q), G〈x; q〉, ZG(x1, x2, x3, . . .), ΓG(x1, x2, x3, . . .), for G = Fw(α) , in terms of Fw. Special instances of our formulas include the exponential formula, Fw(α)(x) = exp(αFw(x)) = (Fw(x)) , cyclotomic identities, and their q-analogues. The virtual weighted species, Λ, is, in fact, a new combinatorial lifting of the function (1 + x).