Discrete interpolation between monotone probability and free probability

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.