# Descent Statistics on Shuffles

Lara Pudwell

## College

Arts and Sciences

## Discipline(s)

Mathematics & Statistics

## ORCID Identifier(s)

0000-0002-7470-6538, 0000-0002-1405-7046, 0000-0002-5330-5946

## Presentation Type

Poster Presentation

Spring 4-23-2016

## Abstract

In mathematics, a permutation is a list of numbers where order of the numbers matters. For many years, the mechanics of shuffling, like a deck of cards, has been studied by many different mathematicians. Our research team examined the question, "What happens if you take two permutations of the same length and shuffle the numbers like you would shuffle a deck of cards?" Doing this produces what we call a shuffle. In 2009, Camillia Smith Barnes counted the number of shuffles of the permutation 12….m with the permutation 12…n, or permutations of different lengths. Our research group refined her work in the following way. A descent in a shuffle is a location where one number is immediately followed by a smaller number. We count the number of shuffles of 12…n and 12…n, or permutations of the same size, with a given number of descents. In our research, we looked to explain a pattern for the number of shuffles of any sized permutation based on descents.