## Symposium on Undergraduate Research and Creative Expression (SOURCE)

#### Title

Exploring a New Object: the Taumutation

Lara Pudwell

#### College

Arts and Sciences

#### Discipline(s)

Mathematics and Statistics

#### Presentation Type

Poster Presentation

Spring 5-1-2020

#### Abstract

We define a taumutation as an nxn grid with exactly two different points in each row and column. A well known mathematical object is the permutation, which is defined as an ordered list of the elements 1,2,3,...,n. Examples of permutations of length 4 include 1423 and 2134. By thinking of the position of an element in a permutation as an x-coordinate and setting its value to be the y-coordinate, we obtain an nxn grid with only one point in each row and column. In a way, a taumutation is two permutations plotted on the same grid. We are often interested in permutations that avoid patterns. For example, permutations that avoid the pattern 132 do not have three elements from left-to-right (not necessarily consecutive), such that the first is the smallest, the second the largest, and the third between them. The space of permutations under pattern avoiding restrictions is well-documented; however, no one has explored our new mathematical object. In our work, we find a way to count how many taumutations exist on an nxn grid when we avoid two permutations of length three within the grid.