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Interval Posets and Polygon Dissections

Eli Bagno, Estrella Eisenberg, Shulamit Reches, Moriah Sigron

TL;DR

The paper addresses the problem of understanding and enumerating interval posets $P(\pi)$ of permutations by establishing a geometric bijection $\Phi$ between interval posets with $n$ minimal elements and diagonally framed dissections of a convex $n+1$-gon without quadrilaterals. The main approach constructs a concrete correspondence: each internal interval $[a,b]$ maps to a diagonal $\{a,b+1\}$, yielding a diagonally framed, quadrilateral-free polygon that encodes the poset structure; this enables a geometric view and direct enumeration. Key contributions include a combinatorial proof that tree interval posets correspond to non-crossing dissections of the $n+1$-gon, and a description of interval posets for block-wise simple permutations as dissections with further restrictions (no triangles or quadrilaterals), along with connections to the OEIS sequence A054514. The framework provides a visual, structural understanding that complements generating-function approaches and has implications for block-wise simple permutation analysis.

Abstract

The Interval poset of a permutation is an effective way of capturing all the intervals of the permutation and the inclusions between them and was introduced recently by Tenner. Thi paper explores the geometric interpretation of interval posets of permutations. We present a bijection between tree interval posets and convex polygons with non-crossing diagonals, offering a novel geometric perspective on this purely combinatorial concept. Additionally, we provide an enumeration of interval posets using this bijection and demonstrate its application to block-wise simple permutations.

Interval Posets and Polygon Dissections

TL;DR

The paper addresses the problem of understanding and enumerating interval posets of permutations by establishing a geometric bijection between interval posets with minimal elements and diagonally framed dissections of a convex -gon without quadrilaterals. The main approach constructs a concrete correspondence: each internal interval maps to a diagonal , yielding a diagonally framed, quadrilateral-free polygon that encodes the poset structure; this enables a geometric view and direct enumeration. Key contributions include a combinatorial proof that tree interval posets correspond to non-crossing dissections of the -gon, and a description of interval posets for block-wise simple permutations as dissections with further restrictions (no triangles or quadrilaterals), along with connections to the OEIS sequence A054514. The framework provides a visual, structural understanding that complements generating-function approaches and has implications for block-wise simple permutation analysis.

Abstract

The Interval poset of a permutation is an effective way of capturing all the intervals of the permutation and the inclusions between them and was introduced recently by Tenner. Thi paper explores the geometric interpretation of interval posets of permutations. We present a bijection between tree interval posets and convex polygons with non-crossing diagonals, offering a novel geometric perspective on this purely combinatorial concept. Additionally, we provide an enumeration of interval posets using this bijection and demonstrate its application to block-wise simple permutations.
Paper Structure (7 sections, 3 theorems, 1 equation, 8 figures)

This paper contains 7 sections, 3 theorems, 1 equation, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Geometrical view of interval posets
  4. General interval posets
  5. Tree interval posets
  6. Interval posets of block-wise simple permutations
  7. Appendix

Key Result

Theorem 3.4

The number of interval posets with $n$ minimal elements is equal to the number of diagonally framed dissections of the convex ($n+1$)-gon such that no quadrilaterals are present (see Figure all in the appendix for some examples of the bijection in small values of $n$).

Figures (8)

  • Figure 1: Interval poset of the permutations: 5123647, 5321647, 4612357, 4632157, 7463215, 7461235, 7532164, 7512364
  • Figure 2: Interval poset of permutations 3142 and 2413.
  • Figure 3: The permutation $\pi=3124576$ and its blocks in a graphical way.
  • Figure 4: Right: the interval poset P. Left: the polygon $\Phi(P)$
  • Figure 5:
  • ...and 3 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Definition 3.6
  • Theorem 3.7
  • proof