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The dimension of the feasible region of pattern densities

Frederik Garbe, Daniel Kral, Alexandru Malekshahian, Raul Penaguiao

TL;DR

We address the problem of determining the degrees of freedom (dimension) in the region of feasible densities of permutation patterns up to size $k$. The main result shows this dimension equals $|{ m P}^L_k|$, the number of non-trivial Lyndon permutations of size at most $k$, revealing a deep link between order structure and density independence. The proof combines permutation limit theory (permutons), Lyndon word theory, and flag-algebra methods to establish matching lower and upper bounds: a Jacobian-based construction yields a local surjectivity for lower bounds, while a universal polynomial representation expresses all densities in terms of Lyndon densities for the upper bound. This resolves a conjecture of Borja BorP20 and highlights Lyndon structure as the governing factor for the degrees of freedom in permutation-pattern densities, with potential implications for other order-sensitive combinatorial limits.

Abstract

A classical result of Erdős, Lovász and Spencer from the late 1970s asserts that the dimension of the feasible region of densities of graphs with at most k vertices in large graphs is equal to the number of non-trivial connected graphs with at most k vertices. Indecomposable permutations play the role of connected graphs in the realm of permutations, and Glebov et al. showed that pattern densities of indecomposable permutations are independent, i.e., the dimension of the feasible region of densities of permutation patterns of size at most k is at least the number of non-trivial indecomposable permutations of size at most k. However, this lower bound is not tight already for k=3. We prove that the dimension of the feasible region of densities of permutation patterns of size at most k is equal to the number of non-trivial Lyndon permutations of size at most k. The proof exploits an interplay between algebra and combinatorics inherent to the study of Lyndon words.

The dimension of the feasible region of pattern densities

TL;DR

We address the problem of determining the degrees of freedom (dimension) in the region of feasible densities of permutation patterns up to size . The main result shows this dimension equals , the number of non-trivial Lyndon permutations of size at most , revealing a deep link between order structure and density independence. The proof combines permutation limit theory (permutons), Lyndon word theory, and flag-algebra methods to establish matching lower and upper bounds: a Jacobian-based construction yields a local surjectivity for lower bounds, while a universal polynomial representation expresses all densities in terms of Lyndon densities for the upper bound. This resolves a conjecture of Borja BorP20 and highlights Lyndon structure as the governing factor for the degrees of freedom in permutation-pattern densities, with potential implications for other order-sensitive combinatorial limits.

Abstract

A classical result of Erdős, Lovász and Spencer from the late 1970s asserts that the dimension of the feasible region of densities of graphs with at most k vertices in large graphs is equal to the number of non-trivial connected graphs with at most k vertices. Indecomposable permutations play the role of connected graphs in the realm of permutations, and Glebov et al. showed that pattern densities of indecomposable permutations are independent, i.e., the dimension of the feasible region of densities of permutation patterns of size at most k is at least the number of non-trivial indecomposable permutations of size at most k. However, this lower bound is not tight already for k=3. We prove that the dimension of the feasible region of densities of permutation patterns of size at most k is equal to the number of non-trivial Lyndon permutations of size at most k. The proof exploits an interplay between algebra and combinatorics inherent to the study of Lyndon words.
Paper Structure (9 sections, 9 theorems, 19 equations, 2 figures)

This paper contains 9 sections, 9 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph limits
  4. Permutations
  5. Permutation limits
  6. Lyndon words and Lyndon permutations
  7. Flag algebra products
  8. Lower bound
  9. Upper bound

Key Result

Theorem 1

For every $k\ge 2$, there exist $x_0\in [0,1]^{{\cal G}^C_k}$ and $\varepsilon>0$ such that for every $x\in B_{\varepsilon}(x_0)\subseteq [0,1]^{{\cal G}^C_k}$, there exists a graphon $W$ such that Moreover, there exists a polynomial function $f:[0,1]^{{\cal G}^C_k}\to [0,1]^{{\cal G}_k}$ such that the following holds for every graphon $W$:

Figures (2)

  • Figure 1: The support of the blow-up of the permutation $42315$ scaled by $0.4$, $0.2$, $0.1$, $0.1$ and $0.2$.
  • Figure 2: The permuton $\Pi^L_3$.

Theorems & Definitions (14)

  • Theorem 1: Erdős, Lovász and Spencer ErdLS79
  • Theorem 2
  • Proposition 3
  • Lemma 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • proof : Proof of Theorem \ref{['thm:lower']}
  • Lemma 7
  • ...and 4 more