  We construct a sequence of states called m-monotone product states which give
a discrete interpolation between the monotone product of states of Muraki and
the free product of states of Avitzour and Voiculescu in free probability. We
derive the associated basic limit theorems and develop the combinatorics based
on non-crossing ordered partitions with monotone order starting from depth m.
The Hilbert space representations of the limit mixed moments in the invariance
principle lead to m-monotone Gaussian operators living in m-monotone Fock
spaces, which are truncations of the free Fock space over the square-integrable
functions on the non-negative real line (m=1 gives the monotone Fock space). A
new type of combinatorics of inner blocks leads to explicit formulas for the
mixed moments of m-monotone Gaussian operators, which are new even in the case
of monotone independent Gaussian operators with arcsine distributions.
