  We study the vacuum distribution, under an appropriate scaling, of a family
of partial sums of nonsymmetric position operators on weakly monotone and
monotone Fock spaces, respectively. We preliminary treat the case of weakly
monotone Fock space, and show that any single operator has the vacuum law
belonging to the free Meixner class. After establishing some relations between
the combinatorics of Motzkin and Riordan paths, we give a recursive formula for
the vacuum moments of the law of any finite sum. Since the operators are
monotone independent, the distribution is the monotone convolution of the free
Meixner law above. We also investigate the asymptotic measure for these sums,
which can be seen as "Poisson type" limit law. It turns out to belong to the
free Meixner class, with an atomic and an absolutely continuous part (w.r.t.
the Lebesgue measure). Finally, we briefly apply analogous considerations to
the case of monotone Fock space.
