Permutation Complexity and the Letter Doubling Map
Steven Widmer
TL;DR
This paper explores the combinatorial complexity of infinite permutations derived from binary words under the letter doubling map, providing bounds and exact formulas for specific classes like Sturmian and Thue-Morse words.
Contribution
It introduces bounds and exact formulas for the permutation complexity of infinite words under letter doubling, focusing on Sturmian and Thue-Morse sequences.
Findings
Upper bound for permutation complexity of general words
Exact complexity formulas for Sturmian words
Exact complexity formulas for Thue-Morse word
Abstract
Given a countable set X (usually taken to be N or Z), an infinite permutation of X is a linear ordering of X. This paper investigates the combinatorial complexity of infinite permutations on N associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula for the complexity is established for the Sturmian words and the Thue-Morse word.
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Taxonomy
Topicssemigroups and automata theory · DNA and Biological Computing · Algorithms and Data Compression