enumeration of weighted paths on a digraph and block hook determinant

TL;DR

This paper investigates the determinants of block hook matrices, revealing a factorization property and providing a combinatorial interpretation through weighted path counting in digraphs.

Contribution

It introduces a novel factorization result for block hook matrix determinants and links it to combinatorial path enumeration in weighted digraphs.

Findings

Determinant of block hook matrices factorizes in a structured way.

Provides a combinatorial interpretation via weighted path counting.

Establishes a connection between matrix determinants and graph theory.

Abstract

In this article, we evaluate determinants of block hook matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.