Pairs of eventually constant maps and nilpotent pairs

TL;DR

This paper generalizes Leinster's bijection between operators and nilpotent pairs to pairs of operators between vector spaces, calculating probabilities of nilpotency and counting related map pairs over finite sets.

Contribution

It extends Leinster's correspondence to pairs of operators between vector spaces and determines probabilities and counts of nilpotent pairs in this broader context.

Findings

Probability that a random pair of operators is nilpotent

Number of eventually constant pairs of maps between finite sets

Connection to spanning trees in bipartite graphs

Abstract

Tom Leinster gave a bijective correspondence between the set of operators on a finite-dimensional vector space $V$ and the set of pairs consisting of a nilpotent operator and a vector in $V$. Over a finite field this bijection implies that the probability that an operator be nilpotent is the reciprocal of the number of vectors in $V$. We generalize this correspondence to pairs of operators between pairs of vector spaces and determine the probability that a random pair of operators be nilpotent. We also determine the set-theoretical counterpart of this construction and compute the number of eventually constant pairs of maps between two finite sets, closely related to the number of spanning trees in a complete bipartite graph.