Partial words with a unique position starting a square

TL;DR

This paper investigates the structure of partial words with a unique starting position for powers, revealing bounds on the number of squares they can contain and providing examples with higher powers.

Contribution

It establishes that partial words over a k-letter alphabet can contain at most k squares starting at the same position, contrasting with full words, and constructs examples with multiple higher powers.

Findings

Partial words over a k-letter alphabet can contain at most k squares starting at the same position.

Full words can contain at most one power starting at a unique position.

Binary partial words can contain three higher powers starting at the same position.

Abstract

We consider partial words with a unique position starting a power. We show that over a $k$ letter alphabet, a partial word with a unique position starting a square can contain at most $k$ squares. This is in contrast to full words which can contain at most one power if a unique position starts a power. For certain higher powers we exhibit binary partial words containing three powers all of which start at the same position.