Revisiting The R\'{e}dei-Berge Symmetric Functions via Matrix Algebra

TL;DR

This paper explores the Rédéi-Berge symmetric functions for directed graphs using matrix algebra, building on prior work to deepen understanding of their algebraic structure and related Hamiltonian path results.

Contribution

It introduces a matrix algebra approach to analyze the Rédéi-Berge symmetric functions, expanding on previous algebraic and combinatorial results.

Findings

Unified the analysis of $U_D$ in power sum and Schur bases

Revisited Hamiltonian path results in digraphs

Extended prior work with new algebraic insights

Abstract

We revisit the R\'{e}dei-Berge symmetric function $\mathcal{U}_D$ for digraphs $D$, a specialization of Chow's path-cycle symmetric function. Through the lens of matrix algebra, we consolidate and expand on the work of Chow, Grinberg and Stanley, and Lass concerning the resolution of $\mathcal{U}_D$ in the power sum and Schur bases. Along the way we also revisit various results on Hamiltonian paths in digraphs.