  In this article, we investigate meandric systems having one shallow side: the
arch configuration on that side has depth at most two. This class of meandric
systems was introduced and extensively examined by I. P. Goulden, A. Nica, and
D. Puder in 2020. Shallow arch configurations are in bijection with the set of
interval partitions. We study meandric systems by using moment-cumulant
transforms for non-crossing and interval partitions, corresponding to the
notions of free and boolean independence, respectively, in non-commutative
probability. We obtain formulas for the generating series of different classes
of meandric systems with one shallow side, by explicitly enumerating the
simpler, irreducible objects. In addition, we propose random matrix models for
the corresponding meandric polynomials, which can be described in the language
of quantum information theory, in particular that of quantum channels.
