  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.
