Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access May 20, 2020

A note on Eulerian numbers and Toeplitz matrices

  • Tian-Xiao He EMAIL logo and Peter J.-S. Shiue
From the journal Special Matrices

Abstract

This note presents a new formula of Eulerian numbers derived from Toeplitz matrices via Riordan array approach.

MSC 2010: 05A15; 65B10; 33C45; 39A70; 41A80

References

[1] A. Böttcher, Albrecht and S. M. Grudsky, Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis, Birkhäuser, 2012.Search in Google Scholar

[2] Ch. A. Charalambides, Enumerative combinatorics, CRC Press Series on Discrete Mathematics and its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2002.Search in Google Scholar

[3] L. Comtet, Advanced Combinatorics-The Art of Finite and Infinite expansions, Dordrecht: Reidel, 1974.10.1007/978-94-010-2196-8Search in Google Scholar

[4] E. Deutsch, L. Ferrari, and S. Rinaldi, Production matrices and Riordan arrays, Ann. Comb. 13 (2009), no. 1, 65–85.Search in Google Scholar

[5] T. X. He, Expression and computation of generalized Stirling numbers, J. Combin. Math. Combin. Comput. 86 (2013), 239–268.Search in Google Scholar

[6] T. X. He, L. C. Hsu, and P. J.-S. Shiue, The Sheffer group and the Riordan group, Discrete Appl. Math.155 (2007), no. 15, 1895–1909.Search in Google Scholar

[7] T. X. He and L. W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra Appl. 507 (2016), 77–9510.1016/j.laa.2016.05.035Search in Google Scholar

[8] L. C. Hsu and P. J.-S. Shiue, On certain summation problems and generalizations of Eulerian polynomials and numbers, Discrete Math. 204 (1999), no. 1-3, 237–247.Search in Google Scholar

[9] T. X. He and R. Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math. 309 (2009), no. 12, 3962–3974.Search in Google Scholar

[10] F. T. Howard, Degenerate weighted Stirling numbers, Discrete Math. 57 (1985), no. 1-2, 45–58.Search in Google Scholar

[11] C. Kelly, An algorithm for sums of integer powers, Math. Maga. 57 (1984), no. 5, 296–297.Search in Google Scholar

[12] M. V. Koutras, Eulerian numbers associated with sequences of polynomials, Fibonacci Quart. 32 (1994), no. 1, 44–57.Search in Google Scholar

[13] J. Quaintance and H. W. Gould, Conbinatorial Identities for Stirling Numbers, World Scientific Publ. Co., Pte. Ltd. Singapore, London, Hong Kong, Tokyo, 2016.10.1142/9821Search in Google Scholar

[14] J. Riordan, Introduction to Combinatorial Analysis, Published by the John Wiley & Sons, Inc., New York, 1958.Search in Google Scholar

[15] L. V. Shapiro, S. Getu, W. J. Woan and L. Woodson, The Riordan group, Discrete Appl. Math. 34(1991), 229–239.10.1016/0166-218X(91)90088-ESearch in Google Scholar

[16] R. Sprugnoli, Riordan arrays and combinatorial sums, Discrete Math., 132 (1994), 267–290.10.1016/0012-365X(92)00570-HSearch in Google Scholar

Received: 2020-01-20
Accepted: 2020-03-31
Published Online: 2020-05-20

© 2020 Tian-Xiao He et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 26.9.2023 from https://www.degruyter.com/document/doi/10.1515/spma-2020-0103/html
Scroll to top button