permeaner

Permeaner is a neologism used in discussions of permutation invariance and averaging. It blends the words permutation and mean, and appears in theoretical contexts rather than established literature.

Definition: The permeaner maps a finite sequence to the arithmetic mean of all distinct permutations. For (x1,...,xn), it equals (1/n!) sum over all permutations π of (x_{π(1)} + ... + x_{π(n)}). This simplifies to the standard mean (x1+...+xn)/n; thus it is permutation-invariant but equivalent to the ordinary mean.

Variants: An approximate permeaner may estimate the value by sampling a subset of permutations. A dynamic permeaner could apply to streaming inputs where the sequence is extended over time, maintaining an order-agnostic running average.

Applications: The term is used mainly to illustrate permutation invariance in teaching and in theoretical discussions. It is discussed in analyses of order-agnostic algorithms and symmetry in statistics.

Status: The term is not part of a formal mathematical lexicon and has no widely accepted formal definition beyond the description above. It appears mostly in informal writings and speculative contexts.

Related concepts include permutation symmetry, mean, order invariance, and ensemble averaging.