Omnibus Sequences, Coupon Collection, and Missing Word Counts

TL;DR

This paper investigates Omnibus Sequences, which contain all possible messages of a certain length as subsequences, exploring their properties and connections to the coupon collector problem.

Contribution

It introduces and analyzes properties of Omnibus Sequences, linking them to classical coupon collector theory, and explores their combinatorial and probabilistic aspects.

Findings

Characterization of Omnibus Sequences

Connections established with coupon collector problem

Insights into missing word counts

Abstract

An {\it Omnibus Sequence} of length $n$ is one that has each possible "message" of length $k$ embedded in it as a subsequence. We study various properties of Omnibus Sequences in this paper, making connections, whenever possible, to the classical coupon collector problem.