Enumeration of polyominoes using Macsyma

AbstractThis paper shows the use of a symbolic language, Macsyma, to obtain new exact or asymptotic results in combinatorics. The examples are taken among polyominoes objects. The main purpose is to show how easy it is to bring some methods into operation in order to obtain new results in enumerative combinatorics

A New Coding for Parallelogram Animals

We describe a new coding for parallelogram animals and Motzkin words. From this new coding, we deduce a relation between the Catalan numbers, Motzkin words and parallelogram animals

Series expansions of the percolation probability for directed square and honeycomb lattices

We have derived long series expansions of the percolation probability for site and bond percolation on directed square and honeycomb lattices. For the square bond problem we have extended the series from 41 terms to 54, for the square site problem from 16 terms to 37, and for the honeycomb bond problem from 13 terms to 36. Analysis of the series clearly shows that the critical exponent $\beta$ is the same for all the problems confirming expectations of universality. For the critical probability and exponent we find in the square bond case, $q_c = 0.3552994\pm 0.0000010$, $\beta = 0.27643\pm 0.00010$, in the square site case $q_c = 0.294515 \pm 0.000005$, $\beta = 0.2763 \pm 0.0003$, and in the honeycomb bond case $q_c = 0.177143 \pm 0.000002$, $\beta = 0.2763 \pm 0.0002$. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent $\Delta = 1$.Comment: LaTex with epsf, 26 pages, 2 figures and 2 tables in Postscript format included (uufiled). LaTeX version of tables also included for the benefit of those without access to PS printers (note that the tables should be printed in landscape mode). Accepted by J. Phys.

How To Use Context Free Grammars And Q-Grammars In Maple

. The purpose of this note is to present a MAPLE library computing on algebraic grammars and q-grammars in a first approach of a language or a q-equation. We list here the main procedures. 1. Introduction The methodology of M.P. Schutzenberger [14] has lead to solve many enumeration problems. It consists in coding by words of an algebraic language the combinatorial objects to be enumerated ; if this language is generated by a context free grammar G, the generating function obtained from G is the solution of the problem. In particular, many applications are in the field of planar maps [4] or polyominoes [7, 8]. An overlook to this method can be found in [8] or [15]. Many problems can not be solved by this methodology and need the introduction of q-calculus, for example if the generating function of the studied objects according some parameters is not algebraic. The q-grammars were introduced by Delest and F'edou [5] in order to get some non algebraic equations by means of attribute gr..

Tree Visualisation and Navigation Clues for Information Visualisation

Information visualisation very often requires good navigation aids on large trees, which represent the underlying abstract information. Using trees for information visualisation requires novel user interface techniques, visual clues, and navigational aids. This paper describes a visual clue for trees as well as an automatic folding (clustering) technique, both based on some mathematical concepts and results in combinatorics. Examples are shown how these techniques can be used, and what the further challenges in this area are

How To Use Context Free Grammars And Q-Grammars In Maple

. The purpose of this note is to present a MAPLE library computing on algebraic grammars and q-grammars in a first approach of a language or a q-equation. We list here the main procedures. 1. Introduction The methodology of M.P. Schutzenberger [14] has lead to solve many enumeration problems. It consists in coding by words of an algebraic language the combinatorial objects to be enumerated ; if this language is generated by a context free grammar G, the generating function obtained from G is the solution of the problem. In particular, many applications are in the field of planar maps [4] or polyominoes [7, 8]. An overlook to this method can be found in [8] or [15]. Many problems can not be solved by this methodology and need the introduction of q-calculus, for example if the generating function of the studied objects according some parameters is not algebraic. The q-grammars were introduced by Delest and F'edou [5] in order to get some non algebraic equations by means of attribute gr..