A note on the determinant of a Toeplitz-Hessenberg matrix

Mircea Merca

Special Matrices (2013)

  • Volume: 1, page 10-16
  • ISSN: 2300-7451

Abstract

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The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.

How to cite

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Mircea Merca. "A note on the determinant of a Toeplitz-Hessenberg matrix." Special Matrices 1 (2013): 10-16. <http://eudml.org/doc/267216>.

@article{MirceaMerca2013,
abstract = {The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.},
author = {Mircea Merca},
journal = {Special Matrices},
keywords = {Toeplitz-Hessenberg matrix; integer partition; multinomial coefficient},
language = {eng},
pages = {10-16},
title = {A note on the determinant of a Toeplitz-Hessenberg matrix},
url = {http://eudml.org/doc/267216},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Mircea Merca
TI - A note on the determinant of a Toeplitz-Hessenberg matrix
JO - Special Matrices
PY - 2013
VL - 1
SP - 10
EP - 16
AB - The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.
LA - eng
KW - Toeplitz-Hessenberg matrix; integer partition; multinomial coefficient
UR - http://eudml.org/doc/267216
ER -

References

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  3. [3] J. M. Bogoya, A. Böttcher, S. M. Grudsky, E. A. Maksimenko, Eigenvectors of Hessenberg Toeplitz matrices and a problem by Dai, Geary, and Kadanoff, Linear Algebra Appl. 436 (2012), 3480–3492. [WoS] 
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  5. [5] X. W. Chang, M. J. Gander, S. Karaa, Asymptotic properties of the QR factorization of banded Hessenberg–Toeplitz matrices. Numer. Linear Algebra Appl. 12 (2005), 659–682. Zbl1164.65332
  6. [6] H. Chen, Bernoulli numbers via determinants, Internat. J. Math. Ed. Sci. Tech. 34 (2003), 291–297. [Crossref] 
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  8. [8] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics. Addison-Wesley, New York, 1994. 
  9. [9] A. Inselberg, On determinants of Toeplitz-Hessenberg matrices arising in power series, J. Math. Anal. Appl. 63 (1978), 347–353. [Crossref] Zbl0379.15003
  10. [10] A. J. E. M. Janssen, Asymptotics of the Perron-Frobenius eigenvalue of nonnegative Hessenberg-Toeplitz matrices, IEEE Trans. Inf. Theor., 35 (1989), 1340–1344. Zbl0696.94007
  11. [11] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Clarendon Press, Oxford, 1995. Zbl0824.05059
  12. [12] P. A. MacMahon, Combinatory analysis. Two volumes (bound as one), Chelsea Publishing Co., New York, 1960. 
  13. [13] P. Mongelli, Combinatorial interpretations of particular evaluations of complete and elementary symmetric functions, Electron. J. Combin. 19 (2012), paper 12, 23 pp. Zbl1243.05040
  14. [14] T. Muir, The theory of determinants in the historical order of development. Four volumes, Dover Publications, New York, 1960. 
  15. [15] R. Van Malderen, Non-recursive expressions for even-index Bernoulli numbers: A remarkable sequence of determinants, arXiv:math/0505437v1, 2005. 

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