A Lattice Path Approach to the Multiplicative Powers of Right-Aligned Binomial Matrices
Pascal’s triangle serves as a fundamental source for constructing a diverse array of unique structural matrices in mathematical literature. This study presents a novel combinatorial methodology for computing the multiplicative powers of “right-aligned binomial matrices”, where the ones from the right boundary of Pascal’s triangle are aligned to the rightmost column of the matrix. Such matrix structures have inherent connections to a broad range of theoretical applications, potentially extending to discrete-time dynamic systems, probability models, and network routing protocols. Currently, while the powers of these matrices are typically evaluated using numerical computation systems, these systems operate via “black-box” algorithms that obscure the underlying structural mechanics and generally preclude symbolic analysis. Furthermore, existing theoretical approaches in the literature rely heavily on complex, indirect recurrence relations. In this work, the computation of matrix powers is approached through lattice path enumeration, specifically utilizing weighted Delannoy paths and Fibonacci numbers. The main theorem yields closed-form symbolic expressions that directly evaluate any arbitrary entry of the matrix power, independent of the exponent’s magnitude and order of matrix, without requiring matrix multiplication. Furthermore, a computational implementation in the Wolfram Language is provided to demonstrate the algorithmic efficiency and practical validity of the proposed explicit formulas.