标题： 新的关于交错符号矩阵的双射证明 显示英文标题

标题： New bijective proofs pertaining to alternating sign matrices

摘要： 交替符号矩阵-下降平面分拆（ASM-DPP）双射问题是双射组合学中最引人入胜的开放问题之一，同时也与可积组合学相关。 签名集和签名双射的概念被用于[费舍尔，I. & 科诺瓦林卡，M.，电子。J。Comb.，27 (2020) 3-35.]来构造$\text{ASM}_n \times \text{DPP}_{n-1}$和$\text{DPP}_n \times \text{ASM}_{n-1}$之间的双射。 在这里，我们将基于我们引入的兼容性概念来构建一种更自然的替代签名双射，该概念用于衡量签名双射的自然程度，这是在那篇论文中提出的交替符号矩阵和移位Gelfand-Tsetlin图之间的双射。 此外，我们给出了一个扩展的交替符号矩阵的精细计数的双射证明，该扩展具有$n+3$统计量，首次在[Fischer, I. & Schreier-Aigner, F., Advances in Mathematics 413 (2023) 108831.]中证明。 显示英文摘要

摘要： The alternating sign matrices-descending plane partitions (ASM-DPP) bijection problem is one of the most intriguing open problems in bijective combinatorics, which is also relevant to integrable combinatorics. The notion of a signed set and a signed bijection is used in [Fischer, I. \& Konvalinka, M., Electron. J. Comb., 27 (2020) 3-35.] to construct a bijection between $\text{ASM}_n \times \text{DPP}_{n-1}$ and $\text{DPP}_n \times \text{ASM}_{n-1}$. Here, we shall construct a more natural alternative to a signed bijection between alternating sign matrices and shifted Gelfand-Tsetlin patterns which is presented in that paper, based on the notion of compatibility which we introduce to measure the naturalness of a signed bijection. In addition, we give a bijective proof for the refined enumeration of an extension of alternating sign matrices with $n+3$ statistics, first proved in [Fischer, I. \& Schreier-Aigner, F., Advances in Mathematics 413 (2023) 108831.].