## Description

A 1324-avoiding (0,1)-matrix is an πΓπ matrix that does not contain the 1324-pattern. Our goal is to find the maximum number of 1βs that an π Γ π 1324-avoiding (0,1)-matrix can contain. We build upon Brualdi and Caoβs recent work, where they characterized the π Γ π 1234-avoiding matrices with the maximum number of 1βs. They found that these matrices can contain up to 3(π + π β 3) 1βs. We originally conjectured that 1324-avoiding matrices must contain at most the same number of 1βs, as is the case with the six patterns formed by permutations of {1,2,3}. However, we have found 1324-avoiding matrices that contain more 1βs than those that are 1234-avoiding, and we provide a conjecture for the maximum number of 1βs that a 1324-avoiding matrix can contain.

## Date of Event

Tuesday, November 28, 2023

## Location

Parker Room 301

#### Included in

1324-Avoiding (0,1)-Matrices

Parker Room 301

A 1324-avoiding (0,1)-matrix is an πΓπ matrix that does not contain the 1324-pattern. Our goal is to find the maximum number of 1βs that an π Γ π 1324-avoiding (0,1)-matrix can contain. We build upon Brualdi and Caoβs recent work, where they characterized the π Γ π 1234-avoiding matrices with the maximum number of 1βs. They found that these matrices can contain up to 3(π + π β 3) 1βs. We originally conjectured that 1324-avoiding matrices must contain at most the same number of 1βs, as is the case with the six patterns formed by permutations of {1,2,3}. However, we have found 1324-avoiding matrices that contain more 1βs than those that are 1234-avoiding, and we provide a conjecture for the maximum number of 1βs that a 1324-avoiding matrix can contain.

## Presenter Bio

Speaker: Megan Bennett

Department of Mathematics Halmos College of Arts and Sciences

Faculty Mentor: Dr. Lei Cao

Department of Mathematics Halmos College of Arts and Sciences