The Hopf monoid on nonnesting supercharacters of pattern groups

We construct supercharacter theories for a collection of unipotent matrix groups and produce a Hopf monoid from the supercharacters. These supercharacter theories are coarser than those defined by Diaconis–Isaacs for algebra groups and have supercharacters and superclasses indexed by nonnesting-labeled set partitions. We compute the supercharacter tables and describe the product and coproduct of the Hopf monoid combinatorially. We also show that this Hopf monoid is free.