Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.
  • [2] Sir Thomas Muir, Contributions to the history of determinants, 1900-1920, Blackie and Son, London, 1930.
  • [3] -, The theory of determinants in the historical order of development, Macmillan, London, 1923.
  • [4] J. W. Neuberger, A quasi-analyticity condition in terms of finite differences, Proc. London Math. Soc. (3) 14 (1964), 245–259. MR 0159914 (28 #3130)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A15

Retrieve articles in all journals with MSC: 15A15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0318178-0
PII: S 0002-9939(1973)0318178-0
Article copyright: © Copyright 1973 American Mathematical Society