Given two nonincreasing integral vectors R and S, with the same sum, we denote by \(\mathcal {A}(R,S)\) the class of all (0,1)-matrices with row sum vector R, and column sum vector S. The Bruhat order and the Secondary Bruhat order on \(\mathcal {A}(R,S)\) are both extensions of the classical Bruhat order on Sn, the symmetric group of degree n. These two partial orders on \(\mathcal {A}(R,S)\) are, in general, different. In this paper we prove that if R = (2,2,…,2) or R = (1,1,…,1), then the Bruhat order and the Secondary Bruhat order on \(\mathcal {A}(R,S)\) coincide.

中文翻译：

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Bruhat 阶和二级 Bruhat 阶重合的 (0,1)-矩阵类

给定两个具有相同和的非递增积分向量 R 和 S，我们用 \(\mathcal {A}(R,S)\) 表示所有 (0,1)-矩阵的类，其中行和向量为 R，列和向量 S。 \(\mathcal {A}(R,S)\) 上的布鲁哈特阶和二级布鲁哈阶都是 Sn 上经典布鲁哈特阶的扩展，Sn 是 n 次对称群。\(\mathcal {A}(R,S)\) 上的这两个偏序通常是不同的。在本文中，我们证明如果 R = (2,2,…,2) 或 R = (1,1,…,1)，那么 \(\mathcal {A}(R ,S)\) 重合。