123-avoiding doubly stochastic matrices

Flag-shaped blockers of 123-avoiding permutation matrices

“…Thus, all minimal blockers are minimum. 123-forcing matrices and blockers of 123-avoiding permutation matrices have been previously investigated in [3,2,1].…”

“…We call these permutation matrices cyclic-Hankel permutation matrices or cyclic-Hankel diagonals. The cyclic-Hankel decomposition is illustrated for n = 6 using letters a, b, c, d, e, f below: Lemma 1.1 (Lemma 2.2 in [3]). The number of 0's in an n × n 123-forcing (0, 1)-matrix is at least n. An n × n 123-forcing (0, 1)-matrix with exactly n 0's contains exactly one 0 from each cyclic-Hankel permutation matrix.…”

“…If the zeros are not contained in one row or one column, L-shaped blockers must contain either the (1, n) or (n, 1) position. In [3], it was shown that all blockers with exact n zeros can be obtained from L-shaped blockers by shifting some zeros along the cyclic-Hankel diagonals. The following examples illustrate these results.…”

“…One of our motivations for studying blockers of 123-avoiding permutation matrices is to gain a better understanding of the polytope generated by 123-avoiding permutation matrices, denoted Ω n (123), whose dimension is (n − 1) 2 . In [4], Brualdi and Cao show that each minimal blocker of n × n 123-avoiding permutation matrices determines a facet of Ω n (123), and a face of Ω n (123) lives in dimension (n − 1) 2 − 1. We seek to learn more about the dimension of the faces of Ω n (123) determined by flag-shaped blocker.…”

“…Theorem 4.7 (Theorem 3.7 in [4]). A minimum blocker of all n × n 123-avoiding permutation matrices determines a facet of the polytope Ω n (123) whose extreme points are the n × n 123-avoiding permutation matrices that intersect the blocker exactly once.…”