\Delta-cumulants in terms of moments

The \Delta-convolution of real probability measures, introduced by Bo.zejko, generalizes both free and boolean convolutions. It is linearized by the \Delta-cumulants, and Yoshida gave a combinatorial formula for moments in terms of \Delta-cumulants, that implicitly defines the latter. It relies on the definition of an appropriate weight on noncrossing partitions. We give here two different expressions for the \Delta-cumulants: the first one is a simple variant of Lagrange inversion formula, and the second one is a combinatorial inversion of Yoshida’s formula involving Schr"oder trees.