On the Evaluation of Rectangular Matrix Permanents: A Symmetric and Combinatorial Analysis

This paper presents a combinatorial perspective on evaluating the permanent for a rectangular matrix. It proves that the permanent can be computed using the permanents of its largest square submatrices. The proof employs a structured combinatorial method and reveals a connection to the subset-sum problem, known as the grid shading problem. Furthermore, this study uncovers an inherent symmetry in the distribution of terms, highlighting structured patterns within permanent computation. This perspective bridges combinatorial principles with matrix theory, offering new insights into their interplay.