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Eventually constant maps for two sets and nilpotent pairs

Weixi Chen, Mee Seong Im, Catherine Lillja, Nicolas Rugo

TL;DR

The paper develops new enumerative and structural connections for nilpotent objects across algebraic settings. It proves that nilpotent matrices over the Boolean semiring are counted by directed acyclic graphs, and it counts eventually constant pairs of set maps via a Cayley-type spanning-tree framework on the complete bipartite graph. It then extends Leinster's approach to pairs of linear maps between two vector spaces, introducing a Fitting-decomposition-based structure and balanced vectors, which yield a bijection with hom-spaces and lead to a clean description of nilpotent pairs over finite fields. Together, these results provide unified counting formulas and structural insights linking nilpotency, graph theory, and linear-algebraic decompositions with potential implications for representation theory and combinatorics.

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.

Eventually constant maps for two sets and nilpotent pairs

TL;DR

The paper develops new enumerative and structural connections for nilpotent objects across algebraic settings. It proves that nilpotent matrices over the Boolean semiring are counted by directed acyclic graphs, and it counts eventually constant pairs of set maps via a Cayley-type spanning-tree framework on the complete bipartite graph. It then extends Leinster's approach to pairs of linear maps between two vector spaces, introducing a Fitting-decomposition-based structure and balanced vectors, which yield a bijection with hom-spaces and lead to a clean description of nilpotent pairs over finite fields. Together, these results provide unified counting formulas and structural insights linking nilpotency, graph theory, and linear-algebraic decompositions with potential implications for representation theory and combinatorics.

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.

Paper Structure

This paper contains 8 sections, 14 theorems, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Nilpotent maps over B
  3. Eventually constant pairs of set-theoretic functions
  4. Introduction to graph theory
  5. Eventually constant pairs of functions
  6. Nilpotent pairs and balanced vectors
  7. Linear algebra for two vector spaces
  8. Nilpotent pairs of vector spaces and balanced vectors

Key Result

Lemma 2.1

Let $A = (A_{ij})\in \mathcal{N}_n(\mathbb{B})$. Then the diagonal coordinates of $A$ all equal 0. If the coordinate $A_{ij}=1$, then $A_{ji}=0$. A Boolean matrix is nilpotent if and only if there is no sequence $i_1$, $i_2$, $\ldots$, $i_k$ satisfying $A_{i_1,i_2}$$=$$A_{i_2,i_3}$$=$$\ldots$$=$$A_{

Figures (5)

  • Figure 1: There are $1$ directed acyclic graph (DAG) on $1$ (ordered) vertex, and $3$ DAGs on $2$ ordered vertices.
  • Figure 2: There are $25$ directed acyclic graphs (DAGs) on $3$ ordered vertices.
  • Figure 3: Left: there is one eventually constant map when $|X|=|Y|=1$. Right: there are two eventually constant maps when $|X|=2$ and $|Y|=1$.
  • Figure 4: The eventually constant maps when $|X|=|Y|=2$.
  • Figure 5: This is an example of a balanced vector $v$ of length $\ell$ and its images under the compositions of the linear maps $f:V\rightarrow W$ and $g:W\rightarrow V$.

Theorems & Definitions (32)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Remark 2.6
  • ...and 22 more