Characterization of the allowed patterns of signed shifts

Abstract

The allowed patterns of a map are those permutations in the same relative order as the initial segments of orbits realized by the map. In this paper, we characterize and provide enumerative bounds for the allowed patterns of signed shifts, a family of maps on infinite words.

Characterizations and enumerations of patterns of signed shifts

“…Similarly, when π n = 1 (so that y and q are defined) and ζ is defined by a valid σ-segmentation ofπ, let t (m) = ζq 2m Ω σ . The following result is a stronger version of [3,Lem 4.6].…”

“…where Ω σ is the largest word in W k under < σ , given in Equation (4). It is shown in [3] that (M σ , <) is order-isomorphic to (W 0 k , < σ ), that is, there is an order-preserving bijection φ : [0, 1] → W 0 k such that φ(M σ (x)) = Σ σ (φ(x)) for all x. As a consequence, M σ has the same allowed patterns as the map Σ σ restricted to W 0 k .…”

“…Ascents and increasing sequences are defined in the same fashion. A variant of the following definition was first given in [3]. Throughout the paper, we assume that σ = σ 0 σ 1 .…”

“…The above characterization of the allowed patterns of signed shifts is simpler than those given in [1] and [3], and it will allow us to obtain a complete description of the words w ∈ W k inducing π in Theorem 4.4, as well as enumeration results for the negative shift in Section 6. We note that the characterization given in [1] is incomplete.…”

“…We note that the characterization given in [1] is incomplete. While this is corrected in [3,Thm. 3.10], the determination of whether π ∈ Allow(Σ σ ) uses an unhandy condition, namely the requirement that no b satisfies • π n−b < π n < π n−2b or π n−2b < π n < π n−b , and • e t < π n−b−i ≤ e t+1 if and only if e t < π n−i ≤ e t+1 for all 1 ≤ i ≤ b and all 0 ≤ t ≤ k. The role of this condition will be played by our notion of an invalid σ-segmentation introduced in Definition 3.5, which is significantly simpler.…”