Combinatorics of NC-Probability Spaces with Independent Constants

Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz. We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension. Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probability theories are closely related to the usual notions. Hence, an important part of the boolean/monotone probability theories can be imported directly. We describe some standard combinatorial aspects of these variations (and their cyclic versions), such as their Mobius functions, which feature well-known combinatorial integer sequences. The new notions with independent constants seem also more strongly related to the operator-valued extension of c-free probability.