关于具有指定行和列权重的近交替符号矩阵的注记
A note on near alternating sign matrices with prescribed row and column weights
AI总结：
本文研究具有指定行和列权重的近交替符号矩阵的存在性，通过图论刻画将其转化为二进制数组的邻接图是否为二部图，并指出Gale--Ryser条件不充分，提出猜想并给出基于凸数组和复合构造的正面结果。
AI中文摘要：
我们研究了具有指定行和列权重的近交替符号矩阵。在回顾基本定义以及由Gale--Ryser定理得出的必要条件之后，我们给出了存在性问题的图论刻画。更确切地说，我们证明了具有指定行和列权重序列的近交替符号矩阵存在，当且仅当存在一个具有相同权重的二进制数组，其关联邻接图是二部图。这种重新表述表明，Gale--Ryser条件在一般情况下并不充分。然后，我们提出了一个关于行和列权重序列置换的自然猜想，并基于凸二进制数组和复合构造，给出了一些正面的结果。
英文摘要：
We study near alternating sign matrices with prescribed row and column weights. After recalling the basic definitions and the necessary conditions coming from the Gale--Ryser theorem, we give a graph-theoretic characterization of the existence problem. More precisely, we show that a near alternating sign matrix with prescribed row and column weight sequences exists if and only if there exists a binary array with the same weights whose associated adjacency graph is bipartite. This reformulation shows that the Gale--Ryser conditions are not sufficient in general. We then formulate a natural conjecture up to permutations of the row and column weight sequences, and present some positive results, based on convex binary arrays and composition constructions.
