Pascal Determinantal Arrays and a Generalization of Rahimpour's Determinantal Identity

TL;DR

This paper introduces Pascal determinantal arrays, generalizes Pascal's triangle through determinants, and proves a symmetry identity that extends Rahimpour's conjecture using algebraic and geometric methods.

Contribution

The paper defines a new family of arrays, Pascal determinantal arrays, and proves a symmetry identity generalizing Rahimpour's conjecture with combined algebraic and geometric proofs.

Findings

Established recursive generation of Pascal determinantal arrays.

Proved the symmetry identity $P^{(k)}_{i,j} = P^{(j)}_{i,k}$ for all indices.

Provided geometric interpretation of array entries as weighted double sticks.

Abstract

We introduce a new infinite family of arrays, the \emph{Pascal determinantal arrays} of order $k$, denoted $PD_k$, which generalize the classical Pascal array via determinantal constructions. We present a recursive algorithm for generating $PD_k$, establish its correctness using Dodgson's condensation and a weighted sliding-cross rule, and provide a geometric interpretation of the entries $P^{(k)}_{i,j}$ as weighted double sticks in the Pascal plane. Our main result proves a conjecture that generalizes Rahimpour's identity: for all $i,j,k \ge 0$, \[ P^{(k)}_{i,j} = P^{(j)}_{i,k}, \] where $P^{(k)}_{i,j}$ is the determinant of the $k \times k$ subarray of the Pascal array starting at $(i,j)$. The proof combines algebraic recurrence techniques with a visual, geometry-based argument that reveals the intrinsic symmetry of determinantal Pascal structures.