On Two-faced Families of Non-commutative Random Variables

We demonstrate that the notions of bi-free independence and combinatorial-hi-free independence of two-faced families are equivalent using a diagrammatic view of hi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any twofaced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.