# k-Spectra of weakly-c-Balanced Words

## TL;DR

This paper studies the properties and possible sizes of the set of length-$k$ scattered factors (k-spectra) of binary words, focusing on balanced and c-balanced words, and explores reconstructing words from these spectra.

## Contribution

It characterizes the cardinalities of k-spectra for balanced and c-balanced binary words and investigates the reconstructability of words from their k-spectra.

## Key findings

- Characterizes possible sizes of k-spectra for balanced words.
- Identifies which cardinalities of k-spectra are obtainable.
- Explores methods for reconstructing words from their k-spectra.

## Abstract

A word $u$ is a scattered factor of $w$ if $u$ can be obtained from $w$ by deleting some of its letters. That is, there exist the (potentially empty) words $u_1,u_2,..., u_n$, and $v_0,v_1,..,v_n$ such that $u = u_1u_2...u_n$ and $w = v_0u_1v_1u_2v_2...u_nv_n$. We consider the set of length-$k$ scattered factors of a given word w, called here $k$-spectrum and denoted $\ScatFact_k(w)$. We prove a series of properties of the sets $\ScatFact_k(w)$ for binary strictly balanced and, respectively, $c$-balanced words $w$, i.e., words over a two-letter alphabet where the number of occurrences of each letter is the same, or, respectively, one letter has $c$-more occurrences than the other. In particular, we consider the question which cardinalities $n= |\ScatFact_k(w)|$ are obtainable, for a positive integer $k$, when $w$ is either a strictly balanced binary word of length $2k$, or a $c$-balanced binary word of length $2k-c$. We also consider the problem of reconstructing words from their $k$-spectra.