  An anyon exclusion statistics, which generalizes the Bose-Einstein and
Fermi-Dirac statistics of bosons and fermions, was proposed by Haldane[1]. The
relevant past studies had considered only anyon systems without any physical
boundary but boundaries often appear in real-life materials. When fusion of
anyons is involved, certain `pseudo-species' anyons appear in the exotic
statistical weights of non-Abelian anyon systems; however, the meaning and
significance of pseudo-species remains an open problem. In this paper, we
propose an extended anyon exclusion statistics on surfaces with gapped
boundaries, introducing mutual exclusion statistics between anyons as well as
the boundary components. Motivated by Refs. [2, 3], we present a formula for
the statistical weight of many-anyon states obeying the proposed statistics. We
develop a systematic basis construction for non-Abelian anyons on any Riemann
surfaces with gapped boundaries. From the basis construction, we have a
standard way to read off a canonical set of statistics parameters and hence
write down the extended statistical weight of the anyon system being studied.
The basis construction reveals the meaning of pseudo-species. A pseudo-species
has different `excitation' modes, each corresponding to an anyon species. The
`excitation' modes of pseudo-species corresponds to good quantum numbers of
subsystems of a non-Abelian anyon system. This is important because often
(e.g., in topological quantum computing) we may be concerned about only the
entanglement between such subsystems.
