The design of an object-oriented framework for combinatorial enumeration is described. Combinatorial enumeration requires a subtle interplay between the search mechanism, the ordering of the space of combinatorial objects, and the symmetries used to prune equivalent cases from the search. EEciency requires that pruning be used as much as possible, while correctness requires the pruning not be u...

A program has been written which recently generated all the (unlabelled) nine-point graphs. Written in MACRO-10 assembly language and run on a 165K PDP10, it generates the complete set of 274,668 graphs in less than six hours. The algorithm on which this program is based is discussed with an emphasis on coding of graphs and various programming techniques designed to save space and time during e...

This is a collection of open problems presented at the Lorentz Workshop “Enumeration Algorithms using Structure” which took place at the Lorentz Center of the University of Leiden (The Netherlands), August 24 30, 2015. The workshop brought together researchers interested in various aspects of enumeration algorithms; in particular classical output-sensitive enumeration including output-polynomia...

We revisit the problem of enumeration of vertex-tricolored planar random triangulations solved in [Nucl. Phys. B 516 [FS] (1998) 543-587] in the light of recent combinatorial developments relating classical planar graph counting problems to the enumeration of decorated trees. We give a direct combinatorial derivation of the associated counting function, involving tricolored trees. This is gener...

We give an elementary proof of interesting combinatorial identity which is particular interest in graph theory and its applications. Two applications to enumeration forests with closed-form expressions are given.

The problem of ranking (or perfect hashing) is well known in Combinatorial Analysis, Computer Science, and Information Theory. There are widely used methods for ranking permutations of numbers {1, 2, ..., n}, n ≥ 1, for ranking binary words of length n with a fixed number of ones and for many other combinatorial problems. Many of these methods have nonexponential memory size and the time of enu...