Generalized Larcombe-Fenessey invariants of matrix powers

TL;DR

This work extends the Larcombe-Fenessey invariance from the classic 2×2 (and tri-diagonal) settings to a broad class of $n\times n$ matrices by developing a weight-theoretic, topological framework. Utilizing weighted digraphs, abstract simplicial complexes, and homology, the authors show that products of entries of powers of $A$ along a closed cycle remain invariant under reversal, provided $A$ is acyclic or satisfies a specific cycle-compatibility condition. The main contributions are a general identity for acyclic matrices and a closely related identity in the presence of certain cycle constraints, each proved via weight-preserving bijections between walk sums. These results generalize prior bijective arguments (Zeilberger) and yield new examples of matrices exhibiting the generalized Larcombe-Fenessey invariance, with potential implications in combinatorics, algebra, and topology.

Abstract

In this article, we have found significant generalization of the invariance properties of powers of matrices discovered by Larcombe, Fenessey and further explored by Zeilberger. Moreover, we found interesting new results exhibiting similar phenomena in a more general setup.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• proof
• proof