English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

 Using the standard Coxeter presentation for the symmetric group $\mathfrak{S}_{n}$, two reduced expressions for the same group element $\textsf{w}$ are said to be commutationally equivalent if one expression can be obtained from the other one  by applying a finite sequence of commutations. The commutation classes can be seen as the vertices of a graph $\widehat{G}(\textsf{w})$, where two classes are connected by an edge if elements of those classes differ by a long braid relation. We compute the radius and diameter of the graph $\widehat{G}(\textsf{w}_{\bf 0})$,  for the longest element  $\textsf{w}_{\bf 0}$ in the symmetric group $\mathfrak{S}_{n}$, and show that it is not a planar graph for $n\geq 6$. We also describe a family of commutation classes which contains all atoms, that is classes with one single element, and a subfamily of commutation classes whose elements are in bijection with standard Young tableaux of certain moon-polyomino shapes.

Commutation Classes of the Reduced Words for the Longest Element of ${\mathfrak S}_n$