Combinary

Combinary is a theoretical framework that combines elements of combinatorics with binary encoding to represent and manipulate finite collections of objects. It provides a compact, index-based representation of combinatorial objects such as subsets, sequences, and partitions, enabling efficient ranking, unranking, and enumeration.

In the standard Combinary encoding, a k-subset of an n-element set is mapped to a unique nonnegative integer called its combinary index. This index is computed using a mixed-base representation based on binomial coefficients, similar in spirit to the combinatorial number system. Conversely, a combinary index can be unranked to recover the original subset. Variants support fixed or variable set sizes, multisets, and ordered collections.

Algorithms associated with Combinary include ranking (computing an index from a given object), unranking (recovering an object from an index), and next-element generation for iterating over related objects. These algorithms typically run in time proportional to the size of the represented object, often O(k) for k-subsets, making Combinary suitable for dense enumeration and neighborhood search tasks.

Variants of the framework accommodate different constraints, such as representing partitions, compositions, or multisets, and can integrate with additional data encoding schemes for combined indexing. The approach also supports operations like range queries over indices and permutation-aware representations, depending on the chosen encoding.

Applications and relation: Combinary concepts appear in enumerative combinatorics, data compression, database indexing, and combinatorial optimization, where compact representation and rapid access to neighboring objects are advantageous. It is closely related to the combinatorial number system and combinadics, providing a binary-encoded perspective on ranking and unranking of subset-based structures.