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

H. Teimoori, H. Khodakarami

TL;DR

The paper extends the Pascal triangle by introducing Pascal determinantal arrays $PD_k$, where each entry $P^{(k)}_{i,j}$ is the determinant of a $k\times k$ block of the classical Pascal array. It proves a symmetric generalization of Rahimpour’s determinantal identity: $P^{(k)}_{i,j}=P^{(j)}_{i,k}$ for all $i,j,k\ge0$, using a blend of a constructive recursive algorithm, Dodgson’s condensation, and a geometric sliding-cross framework. The authors provide a concrete algorithm to generate $PD_k$ from $PD_{k-1}$, justify it with a Dodgson-based recurrence and a weighted cross-invariance, and interpret $P^{(k)}_{i,j}$ as the weight of a double-stick in the Pascal plane. They also connect their findings to known closed forms (Krattenthaler’s formula and Hoggatt binomials), offering a geometric and algebraic synthesis that suggests rich directions for generalized determinantal arrays and combinatorial structures.

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.

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

TL;DR

The paper extends the Pascal triangle by introducing Pascal determinantal arrays , where each entry is the determinant of a block of the classical Pascal array. It proves a symmetric generalization of Rahimpour’s determinantal identity: for all , using a blend of a constructive recursive algorithm, Dodgson’s condensation, and a geometric sliding-cross framework. The authors provide a concrete algorithm to generate from , justify it with a Dodgson-based recurrence and a weighted cross-invariance, and interpret as the weight of a double-stick in the Pascal plane. They also connect their findings to known closed forms (Krattenthaler’s formula and Hoggatt binomials), offering a geometric and algebraic synthesis that suggests rich directions for generalized determinantal arrays and combinatorial structures.

Abstract

We introduce a new infinite family of arrays, the \emph{Pascal determinantal arrays} of order , denoted , which generalize the classical Pascal array via determinantal constructions. We present a recursive algorithm for generating , establish its correctness using Dodgson's condensation and a weighted sliding-cross rule, and provide a geometric interpretation of the entries as weighted double sticks in the Pascal plane. Our main result proves a conjecture that generalizes Rahimpour's identity: for all , where is the determinant of the subarray of the Pascal array starting at . The proof combines algebraic recurrence techniques with a visual, geometry-based argument that reveals the intrinsic symmetry of determinantal Pascal structures.
Paper Structure (9 sections, 1 theorem, 52 equations, 12 figures)

This paper contains 9 sections, 1 theorem, 52 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The Class of Pascal Determinantal Array
  3. From Rahimpour's Identity to a Natural Generalization
  4. A Recursive Algorithm for Pascal Determinantal Arrays
  5. Dodgson's Condensation of Determinants
  6. The Weighted Version of Star of David Rule
  7. The Correctness of the Algorithm
  8. A Geometric Interpretation of $P^{(k)}_{i,j}$
  9. Conclusion and Connections to Prior Work

Key Result

Theorem 1

For the Pascal array $P = (P_{i,j})$,

Figures (12)

  • Figure 1: The Pascal array $P = PD_1$.
  • Figure 2: Pascal determinantal array $PD_2$ (Narayana triangle in square form).
  • Figure 3: Illustration of Conjecture \ref{['HasanConj']}: the determinant of the large square equals that of the small shaded square.
  • Figure 4: A rectangle with the Pascal array entries
  • Figure 5: A cross of size $k$
  • ...and 7 more figures

Theorems & Definitions (3)

  • Theorem 1: Rahimpour's identity m1
  • Conjecture 2: Generalized Rahimpour identity
  • Remark 3