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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A regular determinant of binomial coefficients

Author: Philip C. Tonne
Journal: Proc. Amer. Math. Soc. 41 (1973), 17-23
MSC: Primary 15A15
MathSciNet review: 0318178
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Abstract: Let $ n$ be a positive integer and suppose that each of $ \{ {a_p}\} _1^n$ and $ \{ {c_p}\} _1^n$ is an increasing sequence of nonnegative integers. Let $ M$ be the $ n \times n$ matrix such that $ {M_{ij}} = C({a_i},{c_j})$, where $ C(m,n)$ is the number of combinations of $ m$ objects taken $ n$ at a time. We give an explicit formula for the determinant of $ M$ as a sum of nonnegative quantities. Further, if $ {a_i} \geqq {c_i},i = 1,2, \cdots ,n$, we show that the determinant of $ M$ is positive.

References [Enhancements On Off] (What's this?)

  • [1] S. Günther, Von der expliciten Darstelling der regulären Determinanten aus Binomialcoefficienten, Z. Math. Phys. 24 (1879), 96-103.
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Article copyright: © Copyright 1973 American Mathematical Society