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3D permutations and triangle solitaire

Juliette Schabanel

TL;DR

This work addresses pattern avoidance in $3$-dimensional permutations by constructing an explicit bijection to triangle bases arising from a Ledrappier $TEP$ subshift, solving a conjecture of Bonichon and Morel. The core method defines the map $\\Gamma$ from a $3$-permutation $(\\sigma,\\tau)$ to inversion-based coordinates $(r_\\sigma(i), l_\\tau(i))$, showing the image lies in $T_n$ and that the map is bijective on the specified avoidance class via a recursive shifted-sum decomposition that mirrors a cut operation. A dynamical system, the triangle solitaire, is then extended to $3$-permutations, yielding an orbit correspondence that transfers uniform sampling from bases to pattern-avoiding $3$-permutations. The results connect combinatorial pattern-avoidance and symbolic dynamics, enabling transfer of enumeration, sampling, and structural techniques with potential generalizations to higher dimensions and broader pattern families.

Abstract

We provide a bijection between a class of 3-dimensional pattern avoiding permutations and triangle bases, special sets of integer points arising from the theory of tilings and TEP subshifts. This answers a conjecture of Bonichon and Morel.

3D permutations and triangle solitaire

TL;DR

This work addresses pattern avoidance in -dimensional permutations by constructing an explicit bijection to triangle bases arising from a Ledrappier subshift, solving a conjecture of Bonichon and Morel. The core method defines the map from a -permutation to inversion-based coordinates , showing the image lies in and that the map is bijective on the specified avoidance class via a recursive shifted-sum decomposition that mirrors a cut operation. A dynamical system, the triangle solitaire, is then extended to -permutations, yielding an orbit correspondence that transfers uniform sampling from bases to pattern-avoiding -permutations. The results connect combinatorial pattern-avoidance and symbolic dynamics, enabling transfer of enumeration, sampling, and structural techniques with potential generalizations to higher dimensions and broader pattern families.

Abstract

We provide a bijection between a class of 3-dimensional pattern avoiding permutations and triangle bases, special sets of integer points arising from the theory of tilings and TEP subshifts. This answers a conjecture of Bonichon and Morel.

Paper Structure

This paper contains 23 sections, 17 theorems, 1 equation, 14 figures.

Table of Contents

  1. Introduction
  2. Definitions and Setting
  3. 3-Permutations avoiding a pattern
  4. Permutations avoiding a pattern
  5. Higher dimension permutations
  6. Tilings, configurations and bases
  7. Tilings
  8. Filling
  9. Independence and Bases
  10. The bijection
  11. Inversions
  12. The function $\Gamma$
  13. Sparsity
  14. Cuts and shifted sums
  15. Solitaire
  16. ...and 8 more sections

Key Result

Theorem 1

There is an explicit bijection between $3$-permutations of size $n$ avoiding patterns $(\textcolor{colorz}{12}, \textcolor{colory}{12})$ and $(\textcolor{colorz}{312}, \textcolor{colory}{231})$ and triangle bases of size $n$.

Figures (14)

  • Figure 1: The permutation $\sigma = 324615$ with an occurrence of the pattern $231$ in red.
  • Figure 2: Left : The two forbidden patterns : $(\textcolor{colorz}{12}, \textcolor{colory}{12})$ (left) and $(\textcolor{colorz}{312}, \textcolor{colory}{231})$ (right). Right : a $3$-permutation, $(\textcolor{colorz}{54231}, \textcolor{colory}{32514})$, with an occurrence of $(\textcolor{colorz}{312}, \textcolor{colory}{231})$ in orange
  • Figure 3: An example of a tiling. Left: the rule set $\mathcal{R}$. Middle: a valid tiling. Right: an invalid tiling with a forbidden pattern highlighted.
  • Figure 4: Left: The tiling rules of the XOR automaton. Right: An example of a valid tiling. Adding or removing a pattern from $\mathcal{R}$ breaks the TEP property.
  • Figure 5: An example of application of the filing process. Dark gray cells are the original set, colored ones are added by the current step and light gray cells were filled earlier.
  • ...and 9 more figures

Theorems & Definitions (38)

  • Theorem 1
  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 2: SaSc23
  • Theorem 3: SaSc23
  • Proposition 4
  • proof
  • Example 2
  • Proposition 5
  • ...and 28 more