Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T04:47:35.267Z Has data issue: false hasContentIssue false
  • English
  • Français

Succession rules and Deco polyominoes

Published online by Cambridge University Press:  15 April 2002

Elena Barcucci ,
Sara Brunetti  and
Francesco Del Ristoro
Elena Barcucci
Affiliation:
Dipartimento di Sistemi e Informatica, Via Lombroso 6/17, 50134 Firenze, Italy; (barcucci@dsi.unifi.it)
Sara Brunetti
Affiliation:
Dipartimento di Sistemi e Informatica, Via Lombroso 6/17, 50134 Firenze, Italy; (brunetti@dsi.unifi.it)
Francesco Del Ristoro
Affiliation:
Dipartimento di Sistemi e Informatica, Via Lombroso 6/17, 50134 Firenze, Italy; (fdr@dsi.unifi.it)
Get access

Abstract

In this paper, we examine the class of "deco" polyominoes and the succession rule describing their construction. These polyominoes are enumerated according to their directed height by factorial numbers. By changing some aspects of the "factorial" rule, we obtain some succession rules that describe various "deco" polyomino subclasses. By enumerating the subclasses according to their height and width, we find the following well-known numbers: Stirling numbers of the first and second kind, Narayana and odd index Fibonacci numbers. We wish to point out how the changes made on the original succession rule yield some new succession rules that produce transcendental, algebraic and rational generating functions.

Keywords

Polyominoenumerationsuccession rulegenerating function combinatorial interpretation.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 34 , Issue 1 , January 2000 , pp. 1 - 14
Copyright
© EDP Sciences, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barcucci, E., Del Lungo, A., Pergola, E. and Pinzani, R., ECO: A methodology for the Enumeration of Combinatorial Objects. J. Differ. Equations Appl. 5 (1999) 435-490. CrossRef
Barcucci, E., Del Lungo, A. and Pinzani, R., ``Deco'' polyominoes, permutations and random generation. Theoret. Comput. Sci. 159 (1996) 29-42. CrossRef
M. Bousquet-Mélou, q-énumération de polyominos convexes. Publication du LACIM, No. 9 Montréal (1991).
Bousquet-Mélou, M., A method for enumeration of various classes of column-convex polygons. Discrete Math. 151 (1996) 1-25. CrossRef
Delest, M., Gouyou-Beauchamps, D. and Vauquelin, B., Enumeration of parallelogram polyominoes with given bound and site perimeter. Graphs Combin. 3 (1987) 325-339. CrossRef
Delest, M. and Viennot, X.G., Algebraic languages and polyominoes enumeration. Theoret. Comput. Sci. 34 (1984) 169-206. CrossRef
R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley (1989).
F.K. Hwang and C.L. Mallows, Enumerating Nested and Consecutive Partitions. J. Combin. Theory Ser. A 70 (1995) 323-333.
D.E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms. Addison Wesley, Reading Mass (1968).
Kreweras, G., Joint distributions of three descriptive parameters of bridges, edited by G. Labelle and P. Leroux, Combinatoire Énumérative, Montréal 1985. Springer, Berlin, Lecture Notes in Math. 1234 (1986) 177-191. CrossRef
Narayana, T.W., Sur les treillis formés par les partitions d'un entier. C.R. Acad. Sci. Paris 240 (1955) 1188-1189.
N.J.A. Sloane and S. Plouffe, The encyclopedia of integer sequences. Academic Press (1995).