Combinatorial Generation

Efficient generation of combinatorial objects is a well-researched area. Among them many literature devoted on the combinatorial Gray code approach where the goal is to find a Hamiltonian path or cycle in a representative graph of the corresponding combinatorial class. Another approach namely genealogical tree approach has been recently introduced where the goal is to find a rooted spanning tree in the representative graph. Researchers have also focused on finding general patterns in the generation techniques of combinatorial classes so that common approaches can be applied to a large number of related problems. Here, we propose a unifying framework for combinatorial generation by giving recursive definition of an abstract combinatorial class. The definition can be instantiated to an array of specific combinatorial classes namely n-tuple, combination, integer partition, set partition and binary trees by specifying the framework parameters appropriately. As an illustration, we show the instantiation of the combinatorial class of n-tuples, combinations and balanced parenthesis strings and also give novel constant-time generation algorithm for each of them.