The dimension of the feasible region of pattern densities

TL;DR

This paper determines the exact dimension of the feasible region of permutation pattern densities of size at most k, linking it to the count of non-trivial Lyndon permutations, thus extending classical graph results to permutations.

Contribution

It proves that the dimension equals the number of non-trivial Lyndon permutations, refining previous lower bounds and connecting algebraic and combinatorial methods.

Findings

Dimension equals the number of non-trivial Lyndon permutations.

Established a precise link between permutation pattern densities and Lyndon permutations.

Extended classical graph density results to the permutation setting.

Abstract

A classical result of Erd\H{o}s, Lov\'asz 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.