  Cumulants are a notion that comes from the classical probability theory, they
are an alternative to a notion of moments. We adapt the probabilistic concept
of cumulants to the setup of a linear space equipped with two multiplication
structures. We present an algebraic formula which involves those two
multiplications as a sum of products of cumulants. In our approach, beside
cumulants, we make use of standard combinatorial tools as forests and their
colourings. We also show that the resulting statement can be understood as an
analogue of Leonov--Shiraev's formula. This purely combinatorial presentation
leads to some conclusions about structure constant of Jack characters.
