# A determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata

## TL;DR

This paper establishes a novel connection between Stirling cycle number determinants and the enumeration of unlabeled acyclic single-source automata, using combinatorial bijections and involutions.

## Contribution

It introduces a new combinatorial interpretation of a determinant of Stirling cycle numbers as counting specific automata structures.

## Key findings

- Determinant counts unlabeled acyclic single-source automata
- Bijection between automata and marked lattice paths
- Sign-reversing involution used for evaluation

## Abstract

We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant.