SetXint

SetXint is a method for encoding finite sets of integers into a single natural number. It is used in areas such as combinatorial generation, data serialization, and indexing where a deterministic, bijective mapping between a set and an integer is desirable. The approach emphasizes a reversible transformation: from a set to an integer and back without loss of information.

One canonical construction uses a fixed bijection g: Z -> N to map each element of the set to a nonnegative integer. For a finite set S = {s1, ..., sk} written in increasing order, the setXint value is defined as n = product over i of p_i^{g(si)}, where p_i is the i-th prime (2, 3, 5, 7, ...). Because prime factorization is unique, this encoding is reversible: factor n to recover the exponents, apply g^{-1} to obtain the original elements, and retrieve S. Variants may employ other encodings, such as Cantor pairing or fixed-width bit representations over a universe, but the prime-exponent method is a common example.

Inverse mapping and determinism: Recovering the set from an encoded integer requires prime factorization and applying the inverse bijection g^{-1}. Determinism is ensured by fixing the order of elements and the chosen encoding scheme. The method naturally handles negative integers through the initial g: Z -> N, which assigns distinct natural numbers to negative and nonnegative values.

Applications and limitations: SetXint provides a bijective, compact representation of finite sets, making it useful for compact keys in databases, exact set comparisons, and combinatorial indexing. However, the resulting integers can grow extremely large even for modest input sizes, limiting practical use for large sets or constrained storage. In such cases, alternative encodings like bitsets or probabilistic hashes may be preferred.

See also: Gödel numbering, prime encoding, Cantor pairing function, bitset encodings.