Eventually constant maps for two sets and nilpotent pairs

TL;DR

This paper establishes bijections linking nilpotent matrices, acyclic graphs, and eventually constant maps, providing new enumeration formulas and generalizations of classical combinatorial results.

Contribution

It introduces novel bijections connecting nilpotent pairs, acyclic graphs, and vector space maps, extending known combinatorial principles and formulas.

Findings

Enumerates nilpotent matrices over Boolean semirings via acyclic graphs.

Provides a bijection between eventually constant map pairs and spanning trees in bipartite graphs.

Derives a formula for counting nilpotent pairs of maps between finite-dimensional vector spaces.

Abstract

We give a bijective correspondence between the number of nilpotent matrices over a Boolean semiring and the number of directed acyclic graphs on ordered vertices. We then enumerate pairs of maps between two finite sets whose composites are eventually constant by forming a bijection that relates a pair of such maps with a spanning tree in a complete bipartite graph, and an edge of said tree. This generalizes the main principle of A. Joyal's proof of Cayley's formula. Finally, we generalize T. Leinster's work by considering a pair of finite-dimensional vector spaces and show a bijectivity between a nilpotent pair of maps and a balanced vector with the hom spaces between them. This leads us to an elegant formula for the number of nilpotent pairs.