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Category of non-crossing partitions with blocks of even size
The category of non-crossing partitions with blocks of even size is a Banica-Speicher category of partitions inducing the corepresentation category of the 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 
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 [BanBichColl07].
This name is to be taken literally. A partition 
is said to have blocks of even size if every block of 
has an even number of legs.
- A partition
is said to have blocks of even size if every block of has an even number of legs. - It is said to be non-crossing if there exist no blocks
and 
of with 
and no legs 
and 
such that 
and 
with respect to the cyclic order of . 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 generated by the partition 
.
Counting
See OEIS A001764
Associated easy quantum groups
Via Tannaka-Krein duality for compact quantum groups, the category of all non-crossing partitions with blocks of even size corresponds to the family 
of free hyperoctahedral quantum groups.
References
