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<p>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&#xa0;Tenner (Order <b>39</b>(3), 523–536 <CitationRef CitationID="CR11">2022</CitationRef>). 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&#xa0;Tenner (Order <b>39</b>(3), 523–536 <CitationRef CitationID="CR11">2022</CitationRef>), 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 Möbius function on such posets.</p>

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The Interval Posets of Permutations Seen from the Decomposition Tree Perspective

  • 期刊论文

  • 发表日期：

    2025年1月8日

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  • Mathilde Bouvel,
  • Lapo Cioni,
  • Benjamin Izart

摘要

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 Tenner (Order 39(3), 523–536 2022). 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 Tenner (Order 39(3), 523–536 2022), 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 Möbius function on such posets.