Decomposable functors and the exponential principle, II

TL;DR

This paper introduces a new framework for the exponential principle using functors and natural transformations in multisort species, enabling intrinsic generation of indecomposable objects and simplifying the proof of the exponential formula.

Contribution

It develops a novel setting for the exponential principle with a minimal axiom, applying functorial language to multisort species and illustrating with enumeration of magic squares.

Findings

A single axiom suffices for the exponential formula in this setting

Framework applies to enumeration problems like magic squares

Intrinsic generation of indecomposables enhances combinatorial enumeration

Abstract

We develop a new setting for the exponential principle in the context of multisort species, where indecomposable objects are generated intrinsically instead of being given in advance. Our approach uses the language of functors and natural transformations (composition operators), and we show that, somewhat surprisingly, a single axiom for the composition already suffices to guarantee validity of the exponential formula. We provide various illustrations of our theory, among which are applications to the enumeration of (semi-)magic squares.