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Abstract for
Gilbert Labelle and Pierre Leroux,
An Extension of the Exponential Formula in
Enumerative Combinatorics
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
${F}_{w} = E({F}_{w}^{c})$, where $E$ and $F_w^c$ respectively denote the
species of
{\it sets} and of {\it connected} $F_w${\it-structures}.
Multiplying by $\alpha$ the weight of each $F_w^c$-structure yields
the species ${F}_{{w}^{( \alpha )}} = E({F}_{ \alpha w}^{c})$.
We introduce a ``universal'' virtual weighted species,
$\Lambda ^{(\alpha)}$, such that
$F_{w^{(\alpha)}} = \Lambda^{( \alpha)}\, \circ \, F_w^+$,
where $F_w^+$ denotes the species of non-empty $F_w$-structures.
Using general properties of $\Lambda^{( \alpha)}$ ,
we compute the various enumerative power series
$G(x)$, $\widetilde{G}(x)$, $\overline{G}(x)$,
$G(x;q)$, $G\langle{x;q}\rangle$,
${Z}_{G}(x_1,x_2,x_3,\ldots)$,
${\Gamma }_{G}(x_1,x_2,x_3,\ldots)$,
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 $q$-analogues.
The virtual weighted species, $\Lambda ^{(\alpha)}$, is, in fact,
a new combinatorial lifting of the function ${(1+x)}^{\alpha }$.
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