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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2301.12481 |
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| _version_ | 1866912846663647232 |
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| author | Teimoori, H. Khodakarami, H. |
| author_facet | Teimoori, H. Khodakarami, H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12481 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pascal Determinantal Arrays and a Generalization of Rahimpour's Determinantal Identity Teimoori, H. Khodakarami, H. Combinatorics 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. |
| title | Pascal Determinantal Arrays and a Generalization of Rahimpour's Determinantal Identity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2301.12481 |