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Bijective proof of a conjecture on unit interval posets

Wenjie Fang

Abstract

In a recent preprint, Matherne, Morales and Selover conjectured that two different representations of unit interval posets are related by the famous zeta map in $q,t$-Catalan combinatorics. This conjecture was proved recently by Gélinas, Segovia and Thomas using induction. In this short note, we provide a bijective proof of the same conjecture with a reformulation of the zeta map using left-aligned colored trees, first proposed in the study of parabolic Tamari lattices.

Bijective proof of a conjecture on unit interval posets

Abstract

In a recent preprint, Matherne, Morales and Selover conjectured that two different representations of unit interval posets are related by the famous zeta map in -Catalan combinatorics. This conjecture was proved recently by Gélinas, Segovia and Thomas using induction. In this short note, we provide a bijective proof of the same conjecture with a reformulation of the zeta map using left-aligned colored trees, first proposed in the study of parabolic Tamari lattices.
Paper Structure (5 sections, 9 theorems, 1 equation, 3 figures)

This paper contains 5 sections, 9 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Preliminary
  4. Unit interval posets and plane trees
  5. Unit interval posets and the zeta map

Key Result

Theorem 2.1

A poset $P$ is a unit interval poset if and only if it is $(3+1)$-free and $(2+2)$-free, that is, the order induced on any four elements cannot be a chain of $3$ elements plus an incomparable element, or two disjoint chains each containing $2$ elements.

Figures (3)

  • Figure 1: Example of $\Lambda_{\mathrm{poset}}$ from a unit interval poset defined by a set of unit intervals to a plane tree
  • Figure 2: The zeta map as composition of bijections mediated by trees. The nodes of the same depth of the tree are grouped together. For the Dyck path on the left, the number of north steps on $x = k$ is the number of children of the $k$-th node in the tree, ordered by increasing depth, then from right to left. The one on the right comes from a clockwise contour walk.
  • Figure 3: Example of the bijections $\varphi$ and $\psi$

Theorems & Definitions (16)

  • Theorem 2.1: scott1964measurement
  • Proposition 2.2: poset-counting
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.7: *[Theorem IV]cataland
  • Remark 3.8
  • Remark 3.9
  • Proposition 4.2
  • ...and 6 more