parityto

Parityto is a theoretical construct in combinatorics and coding theory used to generalize the notion of parity to structured objects such as sequences, graphs, and partitions. It assigns to each object a binary label, typically 0/1 or +1/−1, by combining multiple parity-like criteria that reflect the object's arrangement. A common approach defines parityto(O) as the XOR (mod 2 sum) of several component parities, each measuring a parity attribute such as inversions, fixed points, or counts in a prescribed partition. The aim is for parityto to be invariant under a chosen family of transformations, making it a potential invariant in theoretical analysis.

For a simple example, a binary string of length n can yield parityto by comparing the parity of 1s on even positions with the parity of 1s on odd positions, and taking their XOR. Thus parityto captures a two-way parity relationship beyond standard parity.

Applications are mainly in thought experiments, teaching, and investigations into generalized parity invariants and error-detecting schemes. Parityto is not a standard, universally defined concept; different authors define it in ways that suit their problems. See also parity, invariant, parity-check, generalized parity.