  We establish connections between the lattices of non-crossing partitions of
type B introduced by V. Reiner, and the framework of the free probability
theory of D. Voiculescu.
  Lattices of non-crossing partitions (of type A, up to now) have played an
important role in the combinatorics of free probability, primarily via the
non-crossing cumulants of R. Speicher. Here we introduce the concept of {\em
non-crossing cumulant of type B;} the inspiration for its definition is found
by looking at an operation of ``restricted convolution of multiplicative
functions'', studied in parallel for functions on symmetric groups (in type A)
and on hyperoctahedral groups (in type B).
  The non-crossing cumulants of type B live in an appropriate framework of
``non-commutative probability space of type B'', and are closely related to a
type B analogue for the R-transform of Voiculescu (which is the free
probabilistic counterpart of the Fourier transform). By starting from a
condition of ``vanishing of mixed cumulants of type B'', we obtain an analogue
of type B for the concept of free independence for random variables in a
non-commutative probability space.
