## An extension of the exponential formula in enumerative combinatorics.(English)Zbl 0856.05003

Electron. J. Comb. 3, No. 2, Research paper R12, 14 p. (1996); printed version J. Comb. 3, No. 2, 269-282 (1996).
From the authors’ abstract: Let $$\alpha$$ be a formal variable and $$F_w$$ be a weighted species of structures (class of structures closed under weight-preserving isomorphisms) of the form $$F_w = E (F^c_w)$$, where $$E$$ and $$F^c_w$$ respectively denote the species of sets and of connected $$F_w$$-structures. Multiplying by $$\alpha$$ the weight of each $$F^c_w$$-structure yields the species $$F_{w^{(\alpha)}} = E (F^c_{\alpha w})$$. We introduce a ‘universal’ virtual weighted species, $$\bigwedge^{(\alpha)}$$, such that $$F_{w^{(\alpha)}} = \bigwedge^{(\alpha)} \circ F^+_w$$, where $$F^+_w$$ denotes the species of non-empty $$F_w$$-structures. Using general properties of $$\bigwedge^{(\alpha)}$$, we compute the various enumerative power series $$G(x)$$, $$\widetilde G(x)$$, $$\overline G(x)$$, $$G(x;q)$$, $$G\langle x; q \rangle$$, $$Z_G (x_1, x_2, x_3, \dots)$$, $$\Gamma_G (x_1, x_2, x_3, \dots)$$, for $$G = F_{w^{( \alpha)}}$$, in terms of $$F_w$$. Special instances of our formulas include the exponential formula, $$F_{w^{(\alpha)}} (x) = \exp (\alpha F_w (x)) = (F_w (x))^\alpha$$, cyclotomic identities, and their $$q$$-analogous. The virtual weighted species, $$\bigwedge^{(\alpha)}$$, is, in fact, a new combinatorial lifting of the function $$(1 + x)^\alpha$$.

### MSC:

 05A15 Exact enumeration problems, generating functions
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