====== Category of non-crossing partitions with blocks of even size ======

The **category of non-crossing partitions with blocks of even size** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[free hyperoctahedral quantum group|free hyperoctahedral quantum groups]]. 

===== Definition =====
By the **category of non-crossing partitions with blocks of even size** one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose morphism set is the //set of all non-crossing partitions with blocks of even size//. It was introduced by Banica, Bichon and Collins in [(:ref:BanBichColl07)]. 

This name is to be taken literally. A partition  $p\in\Pscr$ is said to have **blocks of even size** if every block of $p$ has an even number of legs.


  * A partition $p\in \Pscr$ is said to have **blocks of even size** if every block of $p$ has an even number of legs.
  * It is said to be **non-crossing** if there exist no blocks $B$ and $B'$ of $p$ with $B\neq B'$ and no legs $i,j\in B$ and $i',j'\in B'$ such that $i\prec i'\prec j$ and $i'\prec j\prec j'$ with respect to the cyclic order of $p$. See also  [[category of all non-crossing partitions]].
  * The name **set of all non-crossing partitions with blocks of even size** is to be taken literally.

A partition with blocks of even size is in particular of even size itself. Moreover, such a partition, if additionally non-crossing, necessarily also has parity-balanced legs. 

===== Canonical generator =====

The category of non-crossing partitions with blocks of even size is the subcategory of $\Pscr$ generated by the partition $\fourpart$.

===== Counting =====

$$\#\Cscr(0,2k)={1\over 2k+1}{3k\choose k}$$

See [[https://oeis.org/A001764|OEIS A001764]]


===== Associated easy quantum groups =====


Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of all non-crossing partitions with blocks of even size corresponds to the family $(H_N^+)_{N\in \N}$ of [[free hyperoctahedral quantum group|free hyperoctahedral quantum groups]].

===== References =====

[( :ref:BanSp09 >>
author:  Banica, Teodor and Speicher, Roland
title:   Liberation of orthogonal Lie groups
year:    2009
journal: Advances in Mathematics
volume:  222
issue:   4
pages:   1461--150
url:     https://doi.org/10.1016/j.aim.2009.06.009
archivePrefix: arXiv
eprint   :0808.2628
)]

[( :ref:BanBichColl07 >>
author   : Banica Teodor and Bichon Julien and Collins Benoit
title    : The hyperoctahedral quantum group
journal  : Journal of the Ramanujan Mathematical Society
year     : 2007
volume   : 22
number   : 4
pages    : 345--384
archivePrefix: arXiv
eprint   :0701859
)]