Skip to main content
Log in

On a Nesbitt–Carlitz Determinant

  • MATHEMATICS
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

A matrix whose component are binomial coefficients and determinant was calculated earlier by L. Carlitz is investigated. It is shown that Carlitz matrix is the result of binomal specialization for dual Jacobi–Trudi determinant presentation of certain Schur function. It leads to another way to calculate Carlitz determinant based upon symmetric function theory. The eigenvalues of Carlitz matrix are shown to be powers of two as well. In order to calculate these eigenvalues the author uses suitable linear operator on the space of polynomials whose degree does not exceed given number. It is shown that in suitable basis matrix of that linear operator has triangular form with powers of two on its diagonal. Main result is generalised from quadratic to cubic case corresponding to a certain matrix, consisted of trinomial coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. J. D. Niblett, “A theorem of Nesbitt,” Am. Math. Mon. 59, 171–174 (1952).

    Article  MathSciNet  Google Scholar 

  2. L. Carlitz, “A determinant,” Am. Math. Mon. 64, 186–188 (1957).

    Article  Google Scholar 

  3. V. Prasolov, Problems and Theorems in Linear Algebra (Nauka, Moscow, 1996; Am. Math. Soc., Providence, RI, 1994), in Ser.: Translations of Mathematical Monographs, Vol. 134.

  4. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed. (Oxford Univ. Press, Oxford, 1995).

    MATH  Google Scholar 

Download references

Funding

The work is supported by Russian Foundation for Basic Research (grant no. 16-01-00750).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. I. Pimenov.

Additional information

Dedicated to Aleksander Ivanovich Generalov with deep gratitude

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pimenov, K.I. On a Nesbitt–Carlitz Determinant. Vestnik St.Petersb. Univ.Math. 53, 64–67 (2020). https://doi.org/10.1134/S1063454120010082

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454120010082

Keywords:

Navigation