The interval posets of permutations seen from the decomposition tree perspective

TL;DR

This paper explores the structure of interval posets of permutations through decomposition trees, providing new proofs, solving open problems, and computing the Möbius function using combinatorial techniques.

Contribution

It introduces a method to derive interval posets from permutation decomposition trees and solves several open problems in the area.

Findings

Provides an alternative proof of existing results

Solves open problems related to enumeration of interval posets

Computes the Möbius function for these posets

Abstract

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.