发表机构
National Institute of Technology, Akashi College（明石国立技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究广义阶梯划分，通过有限主特化公式推导连续阶梯间的最短序列比值，证明非负性条件，并给出乘积公式及特化下的应用。
AI 中文摘要
广义阶梯划分是通过将普通阶梯的每个盒子替换为固定矩形而获得的。在连续的广义阶梯之间，我们确定了以水平条为连续差的最短唯一序列。共轭序列是以垂直条为连续差的最短唯一序列。我们利用参数为$q$和$t$的首一Macdonald多项式的有限主特化公式，推导出沿这两个序列的显式比值。更一般地，对于由包含关系相关的两个划分，我们证明了在去除其单项式因子后，其比值的逐系数非负性迫使$q$的独立性。除了单列序列外，沿水平序列的逐系数非负性等价于由变量数确定的矩形中的补集。在共轭垂直序列中，逐系数非负性恰好出现在最小可能的变量数处，除了从空划分到一列的比值。迭代连续广义阶梯之间的比值产生一个三角乘积公式。每个这样的比值通过一个满足交换、互反和反演恒等式的中心乘积来表达。在Hall-Littlewood特化中，从初始广义阶梯到水平序列中任何划分的比值是单项式乘以枚举矩形中划分的高斯多项式。在Jack极限中，比值允许乘积公式，并且中心乘积的加法类似物满足奇偶性和交换恒等式。在Schur特化且所有变量等于1时，我们获得了半标准杨表计数的比值乘积公式。我们从连续广义阶梯图之间的平移推导出钩子乘积和标准杨表计数的比值。
英文摘要
Generalized staircase partitions are obtained by replacing each box of an ordinary staircase with a fixed rectangle. We determine the unique shortest horizontal-strip sequence between consecutive generalized staircases and its conjugate vertical-strip sequence. For monic Macdonald polynomials, we derive explicit finite principal-specialization ratios along both sequences. For arbitrary nested partitions, coefficientwise nonnegativity after removal of the monomial factor forces independence of $q$. Along the horizontal sequence, this nonnegativity is characterized by rectangular complementation, apart from the one-column case. Along the vertical sequence, it occurs at the smallest admissible number of variables, apart from the initial column case. Endpoint ratios yield a triangular product formula and a centered product with exchange, reciprocity, and inversion identities. Hall-Littlewood, Jack, and Schur specializations give Gaussian-polynomial, finite-product, and tableau formulas, respectively.
Comments30 pages, 4 figures
