# Palindromic Subsequences in Finite Words

## TL;DR

This paper explores palindromic subsequences in finite words, proposing a new conjecture, providing constructions that match the conjectured bounds, and extending the ideas to larger alphabets and linear words.

## Contribution

It introduces a palindromic counterpart to a known conjecture, constructs infinite series of words meeting the bounds, and generalizes the problem to larger alphabets.

## Key findings

- Constructed infinite series of circular words with subsequences of length 2/3 of total
- Extended the conjecture to words over larger alphabets
- Proposed and discussed related conjectures for linear words

## Abstract

In 1999 Lyngs{\o} and Pedersen proposed a conjecture stating that every binary circular word of length $n$ with equal number of zeros and ones has an antipalindromic linear subsequence of length at least $\frac{2}{3}n$. No progress over a trivial $\frac{1}{2}n$ bound has been achieved since then. We suggest a palindromic counterpart to this conjecture and provide a non-trivial infinite series of circular words which prove the upper bound of $\frac{2}{3}n$ for both conjectures at the same time. The construction also works for words over an alphabet of size $k$ and gives rise to a generalization of the conjecture by Lyngs{\o} and Pedersen. Moreover, we discuss some possible strengthenings and weakenings of the named conjectures. We also propose two similar conjectures for linear words and provide some evidences for them.