### A Determinant of Stirling Cycle Numbers Counts Unlabeled Acyclic Single-Source Automata

*David Callan*

#### 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.
We also give a formula for the number of acyclic automata with a
given set of sources.

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