The interval posets of permutations seen from the decomposition tree perspective
Mathilde Bouvel, Lapo Cioni, Benjamin Izart

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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Genome Rearrangement Algorithms · Polynomial and algebraic computation
