Counting the occurrences of generalized patterns in words generated by a morphism

TL;DR

This paper analyzes the frequency of specific classical and generalized patterns within words generated by a particular morphism, focusing on sequences that avoid repetitions, and provides exact counts for these pattern occurrences.

Contribution

It introduces methods to count pattern occurrences in words derived from morphisms, including classical and generalized patterns, with explicit results for a specific nonrepetitive morphism.

Findings

Counted occurrences of classical patterns in morphism-generated words

Derived exact counts for generalized patterns in a specific morphic sequence

Analyzed pattern distribution in nonrepetitive sequences

Abstract

We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a morphism generating a nonrepetitive sequence.