A note on the determinant of a Toeplitz-Hessenberg matrix

Mircea Merca

Special Matrices (2013)

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

Abstract

top
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

top

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

top
  1. [1] G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing, 1976. Zbl0371.10001
  2. [2] J. M. Bogoya, A. Böttcher, S. M. Grudsky, Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices, Oper. Theory Adv. Appl. 220 (2012), 77–95. Zbl06069988
  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] 
  4. [4] A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), 241–259. [Crossref] Zbl0535.05006
  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] 
  7. [7] K.-W. Chen, Inversion of Generating Functions using Determinants, J. Integer Seq. 10 (2007), Article 07.10.5. Zbl1174.11362
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.