  In this work we study the structure and cardinality of maximal sets of
commuting and anticommuting Paulis in the setting of the abelian Pauli group.
We provide necessary and sufficient conditions for anticommuting sets to be
maximal, and present an efficient algorithm for generating anticommuting sets
of maximum size. As a theoretical tool, we introduce commutativity maps, and
study properties of maps associated with elements in the cosets with respect to
anticommuting minimal generating sets. We also derive expressions for the
number of distinct sets of commuting and anticommuting abelian Paulis of a
given size.
