A short proof of the Deutsch-Sagan congruence for connected non crossing graphs

TL;DR

This paper presents a concise proof employing Lagrange inversion for a modulo 3 congruence related to the count of connected noncrossing graphs, simplifying previous complex proofs.

Contribution

The paper introduces a shorter, more elegant proof of a conjectured congruence for connected noncrossing graphs, advancing combinatorial enumeration methods.

Findings

Established a congruence modulo 3 for the number of connected noncrossing graphs

Provided a proof using Lagrange inversion, simplifying prior approaches

Confirmed the conjecture originally posed by Deutsch and Sagan

Abstract

We give a short proof, using Lagrange inversion, of a congruence modulo 3 for the number of connected noncrossing graphs on n vertices that was conjectured by Emeric Deutsch and Bruce Sagan. A more complicated proof had been given earlier by S.-P. Eu, S.-C. Liu, and Y.-N. Yeh.