The coincidence of the Bruhat order and the secondary Bruhat order on $\mathcal{A}(n,k)$

TL;DR

This paper characterizes when the Bruhat order and secondary Bruhat order are identical on the set of n-by-n (0,1)-matrices with fixed row and column sums, revealing a precise combinatorial equivalence condition.

Contribution

It provides a complete characterization of the conditions under which the two orders coincide on (n,k), a class of structured matrices.

Findings

Orders coincide for n  5

Orders coincide when k  1, 2, n-2, n-1, n for n  6

The result clarifies the structure of (n,k) with respect to these orders.

Abstract

Given a positive integer $n$ and a nonnegative integer $k$ with $k\leq n$, we denote by $\mathcal{A}(n,k)$ the class of all $n$-by-$n$ $(0,1)$-matrices with constant row and column sums $k$. In this paper, we show that the Bruhat order and the secondary Bruhat order coincide on $\mathcal{A}(n,k)$ if and only if either $0\leq n\leq 5$ or $k\in\{0,1,2,n-2,n-1,n\}$ with $n\geq 6$.