## Creative Components

#### Degree Type

Creative Component

Spring 2018

Mathematics

Sung Yell Song

Leslie Hogben

Steve Butler

#### Degree(s)

Master of Science (MS)

Mathematics

#### Abstract

The permanent of an $n\times n$ matrix $A=(a_{i j})$ with real entries is defined by the sum $$\sum_{\sigma \in S_n} \prod_{i=1}^{n} a_{i \sigma(i)}$$ where $S_n$ denotes the symmetric group on the $n$-element set $\{1,2,\dots,n\}$. In this creative component we survey some known properties of permanents, calculation of permanents for particular types of matrices and their applications in combinatorics and linear algebra. Then we follow the lines of van Lint's exposition of Egorychev's proof for the van der Waerden's conjecture on the permanents of doubly stochastic matrices. The purpose of this component is to provide elementary proofs of several interesting known facts related to permanents of some special matrices. It is an expository survey paper in nature and reports no new findings.

Program of Study Committee:

Sung Yell Song, Major Professor

Steve Butler

Jonas Hartwig

Leslie Hogben