Counting scattered palindromes in a finite word

TL;DR

This paper studies the properties and bounds of scattered palindromic subwords in finite words, providing characterizations, Fibonacci-based bounds, and conjectures on maximum counts.

Contribution

It introduces a Fibonacci-based upper bound for scattered palindromic subwords and proposes a conjecture on their maximum number in words with multiple distinct letters.

Findings

Characterization of words with minimal scattered palindromic subwords

Upper bound of F_n for scattered palindromic subwords in length n words

Conjecture on maximum scattered palindromic subwords for words with q distinct letters

Abstract

We investigate the scattered palindromic subwords in a finite word. We start by characterizing the words with the least number of scattered palindromic subwords. Then, we give an upper bound for the total number of palindromic subwords in a word of length $n$ in terms of Fibonacci number $F_n$ by proving that at most $F_n$ new scattered palindromic subwords can be created on the concatenation of a letter to a word of length $n-1$. We propose a conjecture on the maximum number of scattered palindromic subwords in a word of length $n$ with $q$ distinct letters. We support the conjecture by showing its validity for words where $q\geq \frac{n}{2}$.