The Frobenius problem for the shuffle operation

TL;DR

This paper characterizes finite sets of words whose iterated shuffle operation results in a co-finite set, providing bounds on the length of words not included in this operation.

Contribution

It introduces a characterization of sets with co-finite iterated shuffle and establishes bounds on excluded word lengths.

Findings

Identifies conditions for the iterated shuffle to be co-finite

Provides bounds on the length of words not in the iterated shuffle

Advances understanding of shuffle operation properties

Abstract

We characterize the finite sets S of words such that that the iterated shuffle of S is co-finite and we give some bounds on the length of a longest word not in the iterated shuffle of S.