  In this paper we continue the study of permutations avoiding the vincular
pattern $1-32-4$ by constructing a generating tree with a single label for
these permutations. This construction finally provides a clearer explanation of
why a certain recursive formula found by Callan actually counts these
permutations, insofar as this formula was originally obtained only as a
consequence of a very intricated bijection with a certain class of ordered
rooted trees. This responds to a theoretical issue already raised by Duchi,
Guerrini and Rinaldi. As a byproduct, we also obtain an algorithm to generate
all these permutations and we refine their enumeration according to a simple
statistic, which is the number of right-to-left maxima to the right of 1.
