  The interval poset of a permutation is the set of intervals of a permutation,
ordered with respect to inclusion. It has been introduced and studied recently
in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of
the decomposition trees of permutations, describing a procedure to obtain the
former from the latter. We then give alternative proofs of some of the results
in [B. Tenner, arXiv:2007.06142], and we solve the open problems that it posed
(and some other enumerative problems) using techniques from symbolic and
analytic combinatorics. Finally, we compute the M\"obius function on such
posets.
