Decomposable functors and the exponential principle
The authors present the construction of a combinatorial theory for a class of enumeration problems yielding an identity of the form \[ 1+ \sum_{n> 0} a_n Z^n/n!= \exp\Biggl(\sum_{n> 0} b_n Z^n/n!\Biggr) \] or \[ 1+ \sum_{n> 0}\sum_{k> 0} a_{nk}Z^n Y^k/n!= \exp\Biggl(Y \sum_{n> 0} b_nZ^n/n!\Biggr). \] These identities come from the problem of counting labeled combinatorial structures that are composed of a finite number of indecomposable substructures. \textit{E. A. Bender} and \textit{J. R. Goldman} [Indiana Univ. Math. J. 20, 753-765 (1971; Zbl 0217.01803)] dealt with the composition of combinatorial structures axiomatically. \textit{H. S. Wilf} [Generatingfunctionology. Second edition (Academic Press, Boston, MA) (1994; Zbl 0831.05001)] advanced the theory by introducing set-theoretic constructions related to the relevant decomposition. The authors here expand on earlier work of theirs to develop this theory using covariant functors on the category of finite sets and bijective maps. They further relate this work to the connection between group actions and finite-index subgroups; the equation \(X^\alpha= X^\beta\) in symmetric semigroups; and cyclic sets. They then find further applications by introducing topos-like categories.
scientific article; zbMATH DE number 1053706
scientific article