The Interval Posets of Permutations Seen from the Decomposition Tree Perspective

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 recentlyin [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective ofthe decomposition trees of permutations, describing a procedure to obtain theformer from the latter. We then give alternative proofs of some of the resultsin [B. Tenner, arXiv:2007.06142], and we solve the open problems that it posed(and some other enumerative problems) using techniques from symbolic andanalytic combinatorics. Finally, we compute the Möbius function on suchposets.