Permutation complexity of the fixed points of some uniform binary morphisms

An infinite permutation is a linear order on the set N. We study the properties of infinite permutations generated by fixed points of some uniform binary morphisms, and find the formula for their complexity.

💡 Research Summary

The paper investigates the factor (or permutation) complexity of infinite permutations that are induced by the fixed points of certain uniform binary morphisms. An infinite permutation is defined by assigning to each natural number i the binary real number Rω(i)=0.ωiωi+1…, where ω is a right‑infinite binary word, and ordering the indices according to the values of these reals. The permutation complexity λ(n) is the number of distinct sub‑permutations of length n that appear in this infinite permutation.

The authors restrict attention to uniform marked binary morphisms ϕ whose blocks all have the same length ℓ. Within this family they single out a class Q consisting of two possible forms: (i) ϕ(0)=X=01ⁿ0x₁, ϕ(1)=Y=10ᵐ1y₀ with n,m≥1, each of the words 1ⁿ and 0ᵐ occurring exactly once in X and Y respectively, and neither X nor Y ending with 1ⁿ⁻¹ or 0ᵐ⁻¹; (ii) ϕ(0)=01ⁿ, ϕ(1)=10ⁿ where n=ℓ−1. The fixed point ω=ϕ∞(0) of any morphism from Q is shown to be “circular”, meaning that any sufficiently long factor has a unique synchronization point, which guarantees a unique decomposition into ℓ‑length blocks.

Key technical lemmas (1‑3) establish that for any two positions i and j that contain the same symbol (0 or 1), the relative order of Rω(i) and Rω(j) is completely determined by the residues i mod ℓ, j mod ℓ and by the types of the blocks (first‑type or second‑type) in which the positions lie. Consequently, the permutation induced by a factor depends only on the pattern of block types and the intra‑block offsets.

The paper then introduces a classification of factors of ω into three families:

• Bad factors – factors that generate at least one pair of equivalent permutations (equivalence means that the only possible difference lies in the relative order of the first and last elements).
• Narrow factors – factors whose ancestor chain first encounters a bad factor uₖ, and for the interpretation of the preceding factor the left‑cut length i+1 exceeds the right‑cut length ℓ−j.
• Wide factors – analogous to narrow factors but with i+1 < ℓ−j.

For each factor u the authors define f(u) as the number of distinct permutations generated by u, and introduce auxiliary integers m_u and n_u that count, respectively, the number of permutations contributed by the “core” part of the ancestor and the extra permutations arising from the bad factor. Lemmas 5‑11 prove monotonicity (f(u) never decreases when moving to an ancestor) and give explicit formulas:

• For a bad factor u, f(u)=m_a+2 n_a.
• For a narrow factor u, f(u)=m_a+n_a.
• For a wide factor u, f(u)=m_a+2 n_a.

Here a denotes the unique ancestor in the set A of short factors (length < L_ω) whose descendant chain eventually reaches u.

Section 6 develops an algorithm to compute the sum Σ_{|u|=n} f(u) for any n. The set A is split into A₁ (bad short factors) and A₂ (the rest). For each a∈A the quantities C_{type}^a(n) (type∈{bad,narrow,wide,all}) count how many length‑n factors have a as the ultimate ancestor of the given type. Using the formulas from the previous lemmas the total sum can be expressed as a linear combination of these counts and the parameters m_a, n_a. The counts satisfy simple linear recurrences in n, expressed via the decomposition n = xℓ + r (0≤r<ℓ).

Sections 7‑8 focus on “special words”, i.e., factors v such that both extensions v0 and v1 appear in ω. Special words have a unique interpretation (h, v′, i, 0). The authors study how the permutations generated by v0 and v1 relate to those generated by their ancestors. They define g(v) as the number of common permutations generated by v0 and v1, and prove recurrences for the total Σ_{|v|=n−1} g(v) using auxiliary quantities k_v, t_v, r_v that count, respectively, common permutations, those that become equivalent after extension, and mixed pairs. Lemmas 12‑19 give the necessary combinatorial relations, and Theorem 2 provides a closed‑form expression for the sum of g(v) over all special words of a given length.

Finally, Section 9 presents the main theorem (the exact statement is omitted in the excerpt). The proof relies on Lemma 20 (a known result stating that two distinct factors of the same length cannot generate the same permutation) together with all the machinery built earlier. The theorem yields an explicit formula for λ(n), the permutation complexity of the infinite permutation δ_ω, in terms of ℓ, the parameters of the morphism, and the counts derived in the previous sections. As a corollary, the authors recover the known complexity of the Thue‑Morse permutation (ϕ(0)=01, ϕ(1)=10) – namely λ_TM(n)=2n+1−2⌊log₂ n⌋ – providing an alternative proof that fits into the general framework.

In summary, the paper delivers a comprehensive combinatorial analysis of permutation complexity for a broad class of uniform binary morphisms. By introducing the notions of bad, narrow, and wide factors, together with a detailed treatment of special words and their ancestor chains, the authors obtain exact counting formulas that generalize previously known results (e.g., for Sturmian and Thue‑Morse words). The work deepens the connection between subword structure of infinite words and the order‑theoretic properties of the permutations they induce, offering new tools for further investigations in symbolic dynamics, combinatorics on words, and permutation pattern theory.