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Complete non-ambiguous trees and associated permutations: new enumerative results

Thomas Selig, Haoyue Zhu

TL;DR

The paper deepens the CNAT–permutation correspondence by linking CNAT counts to acyclic orientation numbers of permutation graphs, enabling resolution of conjectures on how many CNATs a permutation can be associated with. It delivers a new, direct bijection between upper-diagonal CNATs and (n−1)-permutations, preserving meaningful statistics. Through quadrant and pattern characterizations, it derives exact counts for B(n,1), B(n,2), and B(n,3), proves that no permutation has 5 CNATs, and identifies the maximal CNAT count (n−1)!, attained only by the decreasing permutation. The results offer new combinatorial tools and bijections for future exploration of CNATs and their permutation counterparts, and suggest further pattern-avoidance and core-structure angles to study.

Abstract

We study a link between complete non-ambiguous trees (CNATs) and permutations exhibited by Daniel Chen and Sebastian Ohlig in recent work. In this, they associate a certain permutation to the leaves of a CNAT, and show that the number of $n$-permutations that are associated with exactly one CNAT is $2^{n-2}$. We connect this to work by the first author and co-authors linking complete non-ambiguous trees and the acyclic orientation number of the associated permutation graph. This allows us to prove a number of conjectures by Chen and Ohlig on the number of $n$-permutations that are associated with exactly $k$ CNATs for various $k > 1$, via various bijective correspondences between such permutations. We also exhibit a new bijection between $(n-1)$-permutations and CNATs whose permutation is the decreasing permutation $n(n-1)\cdots1$. This bijection maps the left-to-right minima of the permutation to dots on the top row of the corresponding CNAT, and descents of the permutation to empty rows of the CNAT.

Complete non-ambiguous trees and associated permutations: new enumerative results

TL;DR

The paper deepens the CNAT–permutation correspondence by linking CNAT counts to acyclic orientation numbers of permutation graphs, enabling resolution of conjectures on how many CNATs a permutation can be associated with. It delivers a new, direct bijection between upper-diagonal CNATs and (n−1)-permutations, preserving meaningful statistics. Through quadrant and pattern characterizations, it derives exact counts for B(n,1), B(n,2), and B(n,3), proves that no permutation has 5 CNATs, and identifies the maximal CNAT count (n−1)!, attained only by the decreasing permutation. The results offer new combinatorial tools and bijections for future exploration of CNATs and their permutation counterparts, and suggest further pattern-avoidance and core-structure angles to study.

Abstract

We study a link between complete non-ambiguous trees (CNATs) and permutations exhibited by Daniel Chen and Sebastian Ohlig in recent work. In this, they associate a certain permutation to the leaves of a CNAT, and show that the number of -permutations that are associated with exactly one CNAT is . We connect this to work by the first author and co-authors linking complete non-ambiguous trees and the acyclic orientation number of the associated permutation graph. This allows us to prove a number of conjectures by Chen and Ohlig on the number of -permutations that are associated with exactly CNATs for various , via various bijective correspondences between such permutations. We also exhibit a new bijection between -permutations and CNATs whose permutation is the decreasing permutation . This bijection maps the left-to-right minima of the permutation to dots on the top row of the corresponding CNAT, and descents of the permutation to empty rows of the CNAT.
Paper Structure (14 sections, 31 theorems, 6 equations, 17 figures, 1 table)

This paper contains 14 sections, 31 theorems, 6 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs and orientations
  4. Acyclic orientation numbers
  5. Permutations, permutation patterns, and permutation graphs
  6. Complete non-ambiguous trees and associated permutations
  7. Connecting CNATs to the acyclic orientation numbers
  8. Upper-diagonal CNATs and permutations
  9. Counting CNATs according to their associated permutations
  10. Linking permutations with one and two CNATs
  11. Linking permutations with two and three CNATs
  12. Permutations with $5$ CNATs
  13. Permutations with maximal numbers of CNATs
  14. Conclusion and future perspectives

Key Result

Proposition 2.1

Let $G$ be a graph, and $s \in V(G)$ a fixed sink vertex of $G$. Then the number of $s$-rooted acyclic orientations of $G$ does not depend on the choice of sink vertex $s$. We denote this number $a_G$, and call it the acyclic orientation number.

Figures (17)

  • Figure 1: Illustrating the pruning operation: a graph $G$ on the left, and its 2-core $\mathrm{Core}_2\left( \right)[G]$ on the right. The tree branches of $G$ (removed in the pruning) are represented in blue.
  • Figure 2: Plot of the permutation $\pi = 561243$.
  • Figure 3: The permutation graph of $\pi = 561243$. On the left, we draw the edges corresponding to inversions of $\pi$ on its plot. On the right, we re-draw the permutation graph in more readable form.
  • Figure 4: Two examples of NATs. Leaf dots are represented in blue, and internal dots in black. The NAT on the left is complete, while the one on the right is not (the red dot has only one child).
  • Figure 5: Illustrating the top-row decomposition of a labelled CNAT.
  • ...and 12 more figures

Theorems & Definitions (66)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 56 more