#### Title

Pattern Packing in Words

#### Faculty Sponsor

Dr. Lara Pudwell

#### College

Arts and Sciences

#### Department/Program

VERUM

#### ORCID Identifier(s)

0000-0003-2309-5279, 0000-0003-3722-932X, 0000-0001-5075-647X

#### Document Type

Poster Presentation

#### Symposium Date

Summer 7-30-2018

#### Abstract

A word is an ordered list of numbers. Specifically, a permutation is a word without repeated letters, denoted pi. A pattern is a word we look for within other words, denoted with rho. We use superscript r to represent the reverse of a word. In general, permutations are studied in terms of pattern avoidance, that is, which words avoid which patterns. Researchers have discovered several orderly ways to count pattern avoiding-words of the form pi pi and pi pi^r [1,3].

Instead of avoiding patterns, we study pattern packing; that is, we identify words with as many copies of a pattern as possible. This idea was first studied by Burstein, Hasto, and Mansour, whose focus was packing patterns into general words, whereas ours is packing in words of the form pi pi and pi pi^{r}. In particular, given a pattern rho, we consider how many times we can pack rho into words of these forms, what the rho-optimal words look like, and how many rho-optimal words exist for a given length of pi.

#### Recommended Citation

Krull, Julia; Redmon, Eric D.; and Reimer-Berg, Andrew, "Pattern Packing in Words" (2018). *Summer Interdisciplinary Research Symposium*. 26.

https://scholar.valpo.edu/sires/26

## Biographical Information about Author(s)

Julia Krull is a sophomore math major at Millikin University. Andrew Reimer-Berg is a junior math and computer science major from Eastern Mennonite University. Eric Redmon is junior math and computer science major at Lewis University.