Sur la Classification et le Denombrement des Sousgroupes du Groupe Modulaire et de leurs Classes de Conjugaison
Abstract
In this article we give a classification of the subgroups in PSL(2,Z) and of the conjugacy classes of these subgroups by the mean of an combinatorial invariant: some trivalent diagrams (dotted or not). We give explicit formulae enabling to count the number of isomorphism classes of these structures and of some of their variations, as function of the number of their arcs. Until now, the counting of nondotted diagrams was an open problem, for it gives also the number of unrooted combinatorial maps, triangular or general respectively. The article ends with the description of a high performance algorithm to enumerate those structures witch is built upon an unexpected factoring of the cycle index series of the considered combinatorial species.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2007
 DOI:
 10.48550/arXiv.math/0702223
 arXiv:
 arXiv:math/0702223
 Bibcode:
 2007math......2223A
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 Primary 05C25;
 05C30;
 05C85;
 20F05;
 Secondary 20F10;
 05C38;
 20F36
 EPrint:
 32 pages, ~60 figures