The category of partitions with blocks of even size and parity-balanced legs is a Banica-Speicher category of partitions inducing the corepresentation category of the half-liberated hyperoctahedral quantum groups.

By the category of partitions with blocks of even size and parity-balanced legs one denotes the subcategory of the category of all partitions whose morphism class is the set of partitions with blocks of even size and even distances between legs. It was introduced by Banica, Curran and Speicher in [BanCuSp10].

A partition belongs to this set if the following conditions are met:

• has blocks of even size, i.e., every block of has an even number of legs.
• has parity-balanced legs, i.e., for any block of when counting from an arbitrary point of the partition, the number of legs of at even distances from is equal to the number of legs of at odd distances from .
• The name set of partitions with blocks of even size and parity-balanced legs is to be taken literally.

A partition with blocks of even size is in particular of even size itself.

The category of partitions with blocks of even size and parity-balanced legs is the subcategory of generated by the set of partitions.

Via Tannaka-Krein duality for compact quantum groups, the category of partitions with blocks of even size and parity-balanced legs corresponds to the family of half-liberated hyperoctahedral quantum groups.