functionpermerre

Functionpermerre is a term used in some theoretical computer science and mathematical writing to denote a higher-order operation that transforms a function into a family of functions through permutation-related symmetry. The concept is not part of standard vocabulary in mainstream mathematics, but it appears in niche glossaries and pedagogical materials as an example of how symmetry can be manipulated at the function level.

Formally, let S be a finite set and p a permutation of S. For a function f: S -> R, the functionpermerre of f with respect to p is the function f_p: S -> R defined by f_p(x) = f(p(x)). When a set of permutations P is given, one can consider the collection {f_p | p in P}. The operation can thus be viewed as precomposition by a permutation, linking it to the action of the permutation group on the domain of f.

Properties of functionpermerre reflect the group structure of permutations. If p is the identity, f_p = f. If p and q are permutations, then (f_p)_q corresponds to f_(qp), illustrating how successive applications align with permutation composition. The construction is invariant under relabelling of elements of S, in the sense that permuting the labels yields an equivalent family of transformed functions.

Example: with S = {1,2,3}, f(1)=10, f(2)=20, f(3)=30, and p swapping 1 and 2, then f_p(1)=20, f_p(2)=10, f_p(3)=30.

Applications include teaching symmetry concepts, generating symmetric variants of functions for data augmentation, and discussions of equivariance in algorithms. The term’s origin is informal, and functionpermerre remains a niche label rather than a widely adopted concept. See also permutation, symmetry, precomposition, and equivariance.