Enumeration of weighted paths on a digraph and block hook determinant

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.

Keywords

• block hook matrix
• determinants
• weighted path
• combinatorial proof
• 05a10
• 05a05
• 11c20
• 05c30
• 05c38