Decomposable functors and the exponential principle
type: Item
instance of: scholarly article
title: Decomposable functors and the exponential principle
zbMATH Open document ID: 0947.05002
published in: Advances in Mathematics
publication date: 2000-11-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 1053706
zbMATH Keywords: enumeration problems
zbMATH Keywords: identities
zbMATH Keywords: combinatorial structures
zbMATH Keywords: covariant functors
zbMATH Keywords: group actions
zbMATH Keywords: topos-like categories
author: Andreas W. M. Dress
author: T. W. Müller
reviewed by: Gerald L. Alexanderson
MaRDI profile type: Publication
full work available at URL: aima.1997.1648
OpenAlex ID: W2951355421
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DOI: 10.1006/AIMA.1997.1648
description: scientific article; zbMATH DE number 1053706