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for completeness, to go alongside Kendall distance
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(one of the few references that I found so far which do at least consider the “Cayley distance kernel” $\exp(-d_C(-,-,))$ )
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which mentions the Cayley-distance kernel at least in the forword as the “only reasonable bi-invariant distance”.
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I have made explicit (here) the following immediate and elementary but important fact:
Cayley distance is preserved under the canonical inclusions of symmetric groups
$Sym(n) \overset{ i }{ \hookrightarrow } Sym(n+1) \hookrightarrow Sym(n+2) \hookrightarrow \cdots$in that
$d_C( \sigma_1, \sigma_2 ) \; = \; d_C\big( i(\sigma_1), i(\sigma_2) \big) \,.$In other words, when regarding the metric space given by the set of permutations in $Sym(n)$ with their Cayley distance function between them
as an $(\mathbb{R}_{\geq 0}, \geq)$0-enriched category, then the functors induced by the inclusions are fully faithful and hence are full subcategory inclusions.
as a matrix, then the inclusions correspond to principal submatrices.
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