  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.
