A permutation family coherent across lengths: from n to n + 2 its cycle type gains cycles and changes none. The reversal's is: the longer chain's antitone bijection restricted to the middle is the shorter's, with one more pair outside (antitone_cycle_type). A rotation's follows \(\gcd(n, d)\) and is not. Coherence is naturality: the family is a natural transformation of the powers over the chains and their middle inclusions \(n
\hookrightarrow n + 2\) (IsNaturalTransformation, :natural), as it is natural over the letter maps already (power(τ) commutes with it). Sufficient, not necessary: rotate(1) is not coherent, yet its sections' density \(|\Sigma| / |\Sigma|^n\) falls. A struct trait, for a family to register by one specialisation.
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Detailed Description
struct dedekind::category::length_coherent< S >
A permutation family coherent across lengths: from n to n + 2 its cycle type gains cycles and changes none. The reversal's is: the longer chain's antitone bijection restricted to the middle is the shorter's, with one more pair outside (antitone_cycle_type). A rotation's follows \(\gcd(n, d)\) and is not. Coherence is naturality: the family is a natural transformation of the powers over the chains and their middle inclusions \(n
\hookrightarrow n + 2\) (IsNaturalTransformation, :natural), as it is natural over the letter maps already (power(τ) commutes with it). Sufficient, not necessary: rotate(1) is not coherent, yet its sections' density \(|\Sigma| / |\Sigma|^n\) falls. A struct trait, for a family to register by one specialisation.
- Template Parameters
-
S the permutation.
The documentation for this struct was generated from the following file:
- src/main/modules/dedekind/category/periodic.cppm
