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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2512.06414 |
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| _version_ | 1866911398722797568 |
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| author | Faal, Hossein Teimoori Khodakarami, Hasan |
| author_facet | Faal, Hossein Teimoori Khodakarami, Hasan |
| contents | We study the Pascal determinantal arrays $\PD_k$, whose entries $\PD_k(i,j)$ are the $k\times k$ minors of the lower-triangular Pascal matrix $P=( \binom{a}{b} )_{a,b\ge 0}$.
We prove an exact factorization of the row-wise log-concavity operator:
\[
\LC(\PD_k)=\PD_{k-1}\Had\PD_{k+1},
\]
where $\LC(a)_j=a_j^2-a_{j-1}a_{j+1}$ and $\Had$ denotes the Hadamard (entrywise) product.
This identity is established by an elementary algebraic manipulation implicitly based on the idea of start of David rule.
We further prove a general inequality asserting that the log-concavity operator is submultiplicative under Hadamard products of log-concave arrays:
$\LC(A\Had X)\ge\LC(A)\Had\LC(X)$.
Combining the factorization with this inequality yields a uniform algebraic proof that every row of every array $\PD_k$ ($k\ge 1$) is infinitely log-concave, extending the celebrated theorem of
Brändén for the particular case of Pascal's triangle ($\PD_1$) to the entire determinantal hierarchy.
Applications include the log-convexity of $\{\PD_k(i,j)\}_{k\ge 0}$ in the determinantal order $k$ and a family of determinantal Hadamard inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Factorization of the Log-Concavity Operator for Pascal Determinantal Arrays and Their Infinite Row-Wise Log-Concavity Faal, Hossein Teimoori Khodakarami, Hasan Combinatorics We study the Pascal determinantal arrays $\PD_k$, whose entries $\PD_k(i,j)$ are the $k\times k$ minors of the lower-triangular Pascal matrix $P=( \binom{a}{b} )_{a,b\ge 0}$. We prove an exact factorization of the row-wise log-concavity operator: \[ \LC(\PD_k)=\PD_{k-1}\Had\PD_{k+1}, \] where $\LC(a)_j=a_j^2-a_{j-1}a_{j+1}$ and $\Had$ denotes the Hadamard (entrywise) product. This identity is established by an elementary algebraic manipulation implicitly based on the idea of start of David rule. We further prove a general inequality asserting that the log-concavity operator is submultiplicative under Hadamard products of log-concave arrays: $\LC(A\Had X)\ge\LC(A)\Had\LC(X)$. Combining the factorization with this inequality yields a uniform algebraic proof that every row of every array $\PD_k$ ($k\ge 1$) is infinitely log-concave, extending the celebrated theorem of Brändén for the particular case of Pascal's triangle ($\PD_1$) to the entire determinantal hierarchy. Applications include the log-convexity of $\{\PD_k(i,j)\}_{k\ge 0}$ in the determinantal order $k$ and a family of determinantal Hadamard inequalities. |
| title | A Factorization of the Log-Concavity Operator for Pascal Determinantal Arrays and Their Infinite Row-Wise Log-Concavity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.06414 |