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On Some Alternative Characterizations of Riordan Arrays

Published online by Cambridge University Press:  20 November 2018

Donatella Merlini ,
Douglas G. Rogers ,
Renzo Sprugnoli  and
M. Cecilia Verri
Donatella Merlini
Affiliation:
Dipartimento di Sistemi e Informatica, via Lombroso 6/17, 50134 Firenze, Italy e-mail: dada@dsi2.ing.unifi.it
Douglas G. Rogers
Affiliation:
Fernley House, The Green Croxley Green United Kingdom WD3 3HT e-mail: drogers@cs.bgsu.edu
Renzo Sprugnoli
Affiliation:
Dipartimento di Sistemi e Informatica via Lombroso 6/17 50134 Firenze, Italy e-mail: resp@ingfi1.ing.unifi.it
M. Cecilia Verri
Affiliation:
Dipartimento di Sistemi e Informatica via Lombroso 6/17 50134 Firenze, Italy e-mail: verri@ingfi1.ing.unifi.it
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Abstract

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We give several new characterizations of Riordan Arrays, the most important of which is: if {dn,k}n,k∈N is a lower triangular arraywhose generic element dn,k linearly depends on the elements in a well-defined though large area of the array, then {dn,k}n,k∈N is Riordan. We also provide some applications of these characterizations to the lattice path theory.

Keywords

05A1505C38
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 2 , 01 April 1997 , pp. 301 - 320
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Barnabei, M., Brini, A., and Nicoletti, G., Recursive matrices and umbral calculus, J. Algebra 75(1982), 546–573.Google Scholar
2. Fray, R.D. and Roselle, D.P., Weighted lattice paths, Pacific J. Math. 37(1971), 85–96.Google Scholar
3. Fray, R.D., On weighted lattice paths, J. Combin. Theory Ser. A 14(1973), 21–29.Google Scholar
4. Gessel, I.M., A factorization for formal Laurent series and lattice path enumeration, J. Combin. Theory Ser. A 28(1980), 321–337.Google Scholar
5. Goldman, J.R. and Sundquist, T., Lattice path enumeration by formal schema, Adv. in Appl. Math. 13(1992), 216–251.Google Scholar
6. Goodman, E. and Narayana, T.V., Lattice paths with diagonal steps, Canad. Math. Bull. 12(1969), 847–855.Google Scholar
7. Goulden, I.P. and Jackson, D.M., Combinatorial Enumeration, John Wiley & Sons, 1983.Google Scholar
8. Handa, B.R. and Mohanty, S.G., Higher dimensional lattice paths with diagonal steps, Discrete Math. 15(1976), 137–140.Google Scholar
9. Henrici, P., Applied and Computational Complex Analysis, I, Wiley, New York, 1988.Google Scholar
10. Labelle, J., Languages de Dyck généralisés, Ann. Sci. Math. Québec (1) 17(1993), 53–64.Google Scholar
11. Labelle, J., On pairs of non-crossing generalized Dyck paths, J. Statist. Plann. Inference 34(1993), 209–217.Google Scholar
12. Labelle, J., Lattice paths with or without horizontal steps, Congr. Numer. 101(1994), 233–241.Google Scholar
13. Labelle, J. and Yeh, Y., Dyck paths of knight moves, Discrete Appl. Math. 24(1989), 213–221.Google Scholar
14. Labelle, J. and Yeh, Y., Generalized Dyck paths, Discrete Math. 82(1990), 1–6.Google Scholar
15. Merlini, D., Rogers, D.G., Sprugnoli, R., and Verri, M.C., On some alternative characterizations of Riordan arrays, Technical Report 14, Dipartimento di Sistemi e Informatica, Università di Firenze, 1995.Google Scholar
16. Merlini, D., Sprugnoli, R., and Verri, M.C., Algebraic and combinatorial properties of simple, coloured, walks. In: Proceedings of CAAP’94, Lecture Notes in Comp. Sci. 787, 218–233. 1994.Google Scholar
17. Mohanty, S.G. and Handa, B.R., On lattice paths with several diagonal steps, Canad. Math. Bull. 11(1968), 537–545.Google Scholar
18. Moser, L. and Zayachkowski, W., Lattice paths with diagonal steps, Scripta Math. 26(1963), 223–229.Google Scholar
19. Rogers, D.G., Pascal triangles,Catalan numbers and renewal arrays, ,DiscreteMath. 22(1978), 301–310.Google Scholar
20. Rogers, D.G. and Shapiro, L.W., Deques, trees and lattice paths, Lecture Notes in Math. 884(1981), 292–303.Google Scholar
21. Rohatgi, V.K., On lattice paths with diagonals steps, Canad. Math. Bull. 7(1964), 470–472.Google Scholar
22. Shapiro, L., A survey of the Riordan Group, Proc. Amer. Math. Soc., Richmond, Virginia, 1994.Google Scholar
23. Shapiro, L.V., Getu, S., Woan, W.-J., and Woodson, L., The Riordan group, Discrete Appl.Math. 34(1991), 229–239.Google Scholar
24. Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete Math. 132(1994), 267–290.Google Scholar