## Honors Theses

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### Minimum Blockers of 123-Avoiding Permutation Matrices

2023

#### Document Type

Honors Thesis - NSU Access Only

#### Department

Halmos College of Arts and Sciences and the Guy Harvey Oceanographic Research Center

#### Honors College

Farquhar Honors College

#### Honors College Dean

Andrea Nevins, Ph.D.

#### Home College Dean

Holly Lynn Baumgartner, Ph.D.

Lei Cao, Ph.D.

#### Abstract

A blocker of n × n permutation matrices is a set of positions in an n × n matrix that intersects each n × n permutation matrix at least once. A blocker is minimum if removing any position makes it no longer a blocker. The Hankel cyclic decomposition implies that each minimum blocker of 123-avoiding permutation matrices must have a cardinality of at least n, and minimum blockers containing exactly n positions are called minimal blockers. The well-known Frobenius-Konig theorem characterizes the minimal blockers of permutation matrices: any r × s submatrix is a minimal blocker of all permutation matrices if and only if r + s = n + 1. A permutation matrix A is 123-avoiding if it does not contain the 3 × 3 identity matrix as a submatrix. Recently, Brualdi and Cao characterized the minimal blockers of 123 avoiding permutation matrices. They showed that all minimal blockers of 123-avoiding permutation matrices must contain either the (1,n) or (n,1) position and can be obtained by shifting positions from an L-shaped blocker. We continue their study by defining flag shaped blockers, which need not contain the (1,n) nor (n,1) position and may have a cardinality greater than n. We show that all flag-shaped blockers are minimum and all minimal blockers of 123-avoiding permutation matrices are special cases of flag-shaped blocker.

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