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# Encyclopedia of Combinatorial Structures (Introduction)

This Encyclopedia of Combinatorial Structures (ECS) has originally been written by Stéphanie Petit. The present version is an extended port by Alexis Darrasse and Frédéric Chyzak to the system DynaMoW developed by the latter for the DDMF. The ECS ambitions to be seen as a young cousin of Sloane's Encyclopedia of Integer Sequences with an emphasis on sequences that arise in the context of decomposable combinatorial structures. Like the EIS, it is possible to search the database by the first terms in the sequence, which are then viewed as the sequence of numbers of objects of increasing size in a specified combinatorial structure. It is also possible to search the database by keyword, generating function, or closed form.

The restriction to sequences arising in this combinatorial framework has for advantage that in many cases a lot of information can be computed automatically. Thus, the result of a successful search is a list of combinatorial structures with, for each of them:

• Its name;
• Its combstruct grammar specification;
• A sequence of integers: the (n+1)th term of this sequence is the number of objects of size n defined by the specification. This sequence is computed by the Maple function combstruct[count], which you can use to compute more terms;
• The generating function of this sequence. When the objects are labelled, exponential generating functions are produced. In the unlabelled universe, ordinary generating functions are used. This generating function is obtained with combstruct[gfsolve];
• A linear recurrence for f(n), the number of objects of size n. In order to obtain this recurrence, it is necessary that the generating function be holonomic. This recurrence is computed by gfun[holexprtodiffeq] and gfun[diffeqtorec];
• The closed form for these numbers f(n) (computed either by Maple's rsolve or by gfun[ratpolytocoeff]);
• The first term of the asymptotic expansion of f(n) or f(n)/n! as n tends to infinity. If the objects are unlabelled (ordinary generating functions), these coefficients are the number of objects, otherwise, in the labelled case (exponential generating functions), they are the number of objects divided by n!. This asymptotic behaviour is computed by equivalent which you can use to compute more terms of the expansion;
• A description of the combinatorial structure;
• Some references. When the sequence (f(n)) is in Sloane's Encyclopedia of Integer Sequences, the references contain "EIS nb" with nb the sequence number in the EIS. A reference can also contain the address (URL) of a Web page.

Most of the entries in this list are generated automatically. In some cases, not all the entries could be found by programs and some of them are missing.

This software bases heavily on symbolic libraries by M. Mishna, E. Murray, B. Salvy, and P. Zimmermann. See the links.