Revisiting The Rédei-Berge Symmetric Functions via Matrix Algebra

TL;DR

This work reexamines the Rédei-Berge symmetric function $\mathcal{U}_D$ through a matrix-algebra lens that unifies Chow's path-cycle framework with classical walk-generating techniques. It furnishes explicit determinant-based expressions for $\mathcal{U}_D$ in terms of the adjacency matrix $A$ and its complement $\overline{A}$, and develops comprehensive expansions in both power sums and Schur functions, including $p$-positivity and Schur-positivity results for key graph classes such as acyclic digraphs and tournaments. The paper also extends the framework to Chow's symmetric function $\Xi_D$, preserves established symmetries (e.g., $\mathcal{U}_{\overline{D}}=\omega(\mathcal{U}_D)$ and $\mathcal{U}_{D^{op}}=\mathcal{U}_D$), and links Hamiltonian-path enumeration with immanant-based Schur expansions. Overall, it consolidates and generalizes prior results of Chow, Grinberg–Stanley, and Lass, providing a unified, algebraic toolkit for analyzing Hamiltonian-path phenomena via symmetric-function theory.

Abstract

We revisit the Rédei-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.

Figures (1)

• Figure 1: A digraph with $D$ and its complement $\overline{D}$.

Theorems & Definitions (53)

• Lemma 1
• proof
• Proposition 2
• proof
• Corollary 3
• proof
• Corollary 4
• Theorem 5
• proof
• Proposition 6