Free magmas are graded by a length function. The growth of a submagma is the sequence of the ball sizes using the length as a norm. Given a pair of submagmas N and M, with \(N \subseteq M\) , of a free magma, we define the density of N respect to M as the asymptotic ratio of the growths of N and M. This notion can be interpreted as a generalization of the index’s inverse for groups or as the probability of an element belonging to a submagma. We study the growth and density of several submagmas of a free magma. It is conjectured that finitely generated proper submagmas of the cyclic free magma have null density. In addition, some aspects of enumeration related to the Motzkin paths are shown.
