RutSequenz

RutSequenz is a theoretical construct in combinatorics on words and sequence dynamics. It denotes the sequence produced when a seed sequence is repeatedly transformed by a fixed Rut operator R. The operator is defined by a substitution rule on a finite alphabet and may include a grouping or truncation constraint that governs how outputs are concatenated at each step.

Construction and example: Let A = {0,1}. Define a simple Rut rule by the morphism 0 -> 01 and 1 -> 0. Starting from seed s0 = 0, the RutSequenz is the infinite sequence of iterates s0, R(s0), R(R(s0)), ... which yields 0, 01, 010, 01001, 01001010, 0100101001001, ... This particular instance is closely related to the Fibonacci word, a well-studied morphic sequence. Other rules yield different deterministic sequences with varying growth rates and complexity.

Properties: RutSequenz sequences are deterministic, since they are generated by a fixed rule. Depending on the morphism, the resulting infinite word can be purely morphic, eventually periodic, or a fixed point of the rule. Subword complexity, factor distribution, and associated dynamical systems (via the shift map) are common topics in their study. The concept is often used to illustrate how simple local rules produce global structure.

Origins and usage: The term RutSequenz is used primarily in educational and puzzle contexts to describe a sequence generated by a rule-based operator. It is not a standard term in all literatures but serves as a convenient shorthand for discussions of morphic sequences and substitution systems.

See also: Morphic word, Fibonacci word, Sturmian word, combinatorics on words.