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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor completeness, to go alongside *[[Kendall distance]]* <a href="https://ncatlab.org/nlab/revision/Cayley+distance/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Cayley+distance">current</a>

   for completeness, to go alongside Kendall distance

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 18th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement and proof of Cayley's formula $$ d_C(\sigma_1, \sigma_2) \;=\; n - \left\vert Cycles\big( \sigma_1 \circ \sigma_2^{-1} \big) \right\vert $$ <a href="https://ncatlab.org/nlab/revision/diff/Cayley+distance/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+distance/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Cayley+distance">current</a>

   added statement and proof of Cayley’s formula

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 18th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[M. A. Fligner]], [[J. S. Verducci]], Section 4 of: *Distance Based Ranking Models*, Journal of the Royal Statistical Society. Series B (Methodological) Vol. 48, No. 3 (1986), pp. 359-369 ([jstor:2345433](https://www.jstor.org/stable/2345433)) (one of the few references that I found so far which do at least consider the "Cayley distance kernel" $\exp(-d_C(-,-,))$ ) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+distance/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+distance/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Cayley+distance">current</a>

   added pointer to:

   • M. A. Fligner, J. S. Verducci, Section 4 of: Distance Based Ranking Models, Journal of the Royal Statistical Society. Series B (Methodological) Vol. 48, No. 3 (1986), pp. 359-369 (jstor:2345433)

   (one of the few references that I found so far which do at least consider the “Cayley distance kernel” )

   diff, v2, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 18th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexalso added pointer to * [[Michael A. Fligner]], [[Joseph S. Verducci]] (eds.), *Probability Models and Statistical Analyses for Ranking Data* ([doi:10.1007/978-1-4612-2738-0](https://link.springer.com/book/10.1007/978-1-4612-2738-0)) which mentions the Cayley-distance kernel at least in the forword as the "only reasonable bi-invariant distance". <a href="https://ncatlab.org/nlab/revision/diff/Cayley+distance/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+distance/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Cayley+distance">current</a>

   also added pointer to

   • Michael A. Fligner, Joseph S. Verducci (eds.), Probability Models and Statistical Analyses for Ranking Data (doi:10.1007/978-1-4612-2738-0)

   which mentions the Cayley-distance kernel at least in the forword as the “only reasonable bi-invariant distance”.

   diff, v3, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2021
   • (edited Apr 19th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Persi Diaconis]], [[Phil Hanlon]], Section 4 of: *Eigen Analysis for Some Examples of the Metropolis Algorithm*, in Donald Richards (ed.) *Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications*, Contemporary Mathematics Vol. 138, AMS 1992 ([doi:10.1090/conm/138](http://dx.doi.org/10.1090/conm/138), [EFS NSF 392](https://statistics.stanford.edu/research/eigen-analysis-some-examples-metropolis-algorithm), [pdf](https://statweb.stanford.edu/~cgates/PERSI/papers/eigen-metalg.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+distance/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+distance/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Cayley+distance">current</a>

   added pointer to:

   • Persi Diaconis, Phil Hanlon, Section 4 of: Eigen Analysis for Some Examples of the Metropolis Algorithm, in Donald Richards (ed.) Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, Contemporary Mathematics Vol. 138, AMS 1992 (doi:10.1090/conm/138, EFS NSF 392, pdf)

   diff, v4, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * Thaynara Arielly de Lima; Mauricio Ayala-Rincón, *Complexity of Cayley distance and other general metrics on permutation groups* ([doi:10.1109/ColombianCC.2012.6398020](https://doi.org/10.1109/ColombianCC.2012.6398020), [pdf](https://www.mat.unb.br/ayala/shortw4v2.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+distance/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+distance/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Cayley+distance">current</a>

   added pointer to:

   • Thaynara Arielly de Lima; Mauricio Ayala-Rincón, Complexity of Cayley distance and other general metrics on permutation groups (doi:10.1109/ColombianCC.2012.6398020, pdf)

   diff, v4, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 20th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the example ([here](https://ncatlab.org/nlab/show/Cayley+distance#CayleyDistanceOnSym3)) of the Cayley distance function on $Sym(3)$ <a href="https://ncatlab.org/nlab/revision/diff/Cayley+distance/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+distance/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Cayley+distance">current</a>

   added the example (here) of the Cayley distance function on

   diff, v6, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 24th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have made explicit ([here](https://ncatlab.org/nlab/show/Cayley+distance#CayleyDistancePreservedByInclusionOfSymmetricGroups)) 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 functor|fully faithful]] and hence are [[full subcategory]] inclusions. * as a [[matrix]], then the inclusions correspond to [[principal submatrices]]. <a href="https://ncatlab.org/nlab/revision/diff/Cayley+distance/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+distance/14">v14</a>, <a href="https://ncatlab.org/nlab/show/Cayley+distance">current</a>

   I have made explicit (here) the following immediate and elementary but important fact:

   Cayley distance is preserved under the canonical inclusions of symmetric groups

   in that

   In other words, when regarding the metric space given by the set of permutations in with their Cayley distance function between them

   • as an 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.

   diff, v14, current