Binomial determinants: some closed formulae
Laura Gonz\'alez, Francesc Planas-Vilanova
TL;DR
This paper derives closed-form formulas for certain binomial determinants and reestablishes their positivity using alternative proofs, contributing to combinatorial matrix theory.
Contribution
Provides new closed-form expressions for binomial determinants and offers a different proof of their positivity, enhancing understanding of their properties.
Findings
Closed formulas for binomial determinants with consecutive rows or columns
Calculation of generators for left nullspaces of binomial matrices
Alternative proof of the positivity of binomial determinants
Abstract
This paper is intended to give closed formulae for binomial determinants with consecutive or almost consecutive rows or columns, as well as calculating the generator of left nullspaces defined by some binomial matrices. In the meantime, we reprove, by different means, the positivity of binomial determinants shown by Gessel and Viennot.
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