Super-d-complexity of finite words

TL;DR

This paper introduces super-d-complexity, a new measure for finite words based on special scattered subwords with gaps of at least (d-1), providing methods to compute it for rainbow words and analyzing its maximum for general words.

Contribution

It defines super-d-subwords and develops recursive, formulaic, and graph-based methods to compute super-d-complexity, also exploring its maximum in general words.

Findings

Methods to compute super-d-complexity for rainbow words

Formulas and algorithms for super-d-complexity calculation

Analysis of maximum super-d-complexity for words of length n

Abstract

In this paper we introduce and study a new complexity measure for finite words. For positive integer $d$ special scattered subwords, called super-$d$-subwords, in which the gaps are of length at least $(d-1)$, are defined. We give methods to compute super-$d$-complexity (the total number of different super-$d$-subwords) in the case of rainbow words (with pairwise different letters) by recursive algorithms, by mahematical formulas and by graph algorithms. In the case of general words, with letters from a given alphabet without any restriction, the problem of the maximum value of the super-$d$-complexity of all words of length $n$ is presented.