形状为$[[n,n],[n,1]]$的立体标准杨表的Zeilberger递推关系的一个证明
Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics, with an Account of the Human-Machine Methodology Employed
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中文总结 AI 辅助
本文证明了Zeilberger提出的形状为$[[n,n],[n,1]]$的立体标准杨表数量满足的二阶线性递推关系，通过双射结合代数核方法完成证明并得到多项独立计数结果。
中文摘要 AI 辅助
设$g(n)$表示两层形状$[[n,n],[n,1]]$的立体标准杨表（solid standard Young tableaux，solid-SYT）的数量。Zeilberger在其立体标准杨表项目配套的首个严格挑战中，通过经验观察发现$g(n)$满足一个阶数为2、多项式系数次数为12的线性递推关系，并为该递推关系的证明设立了奖项。本文证明了该递推关系。证明过程通过删除-插入双射将$g(n)$简化为四分之一平面内Kreweras型格路的加权计数，我们通过Bousquet-Mélou和Mishna的代数核方法以闭式形式计算了这些格路。在此过程中，我们得到了几项具有独立意义的计数结果：终止于对角线上的反向Kreweras格路的闭式、对角线的显式代数生成函数，以及长度为$3n+1$且终止于$(1,0)$的反向Kreweras格路的选票加权和计数的恒等式。该递推关系本身可从结构上解释：$g$位于$\boldsymbol{Q}(n)$上的2维模中，由两个超几何项张成，这必然产生二阶递推，并通过克莱姆法则得到其系数。
英文摘要
We report solutions to five challenge problems posed by Doron Zeilberger and his collaborators, together with substantial partial progress on two further problems, and we describe the method by which they were obtained. The solved problems are: the Second Computational Chomp Challenge of Ekhad and Zeilberger, for which we exhibit a bar with three winning opening moves; the third challenge of Spahn and Zeilberger, asking whether the restricted permutation counts a_{r,s} and b_{r,s} are holonomic for all r,s>1, answered affirmatively; the First Rigorous Solid Standard Young Tableaux Challenge, for which we prove the conjectured second-order recurrence; the five-dimensional Geode Challenge of Amdeberhan, Kauers and Zeilberger; and Conjectures 2a and 2b of Kauers and Zeilberger, which we obtain from a local limit theorem for excursions of Markov-modulated random walks in cones. Several results of independent interest arise along the way: a staircase theorem constraining the winning opening moves of any Chomp bar, together with a parity theorem for square bars; an explicit algebraic generating function for reverse-Kreweras diagonal walks and a closed form for their diagonal-endpoint counts; a one-dimensional integral representation for diagonal Geode coefficients; and the identity that each Kauers-Zeilberger constant is a universal factor times the square of the apex value of a discrete cone-harmonic function. All of the work reported here was carried out in collaboration with a large language model. The paper sets out the division of labour, records the verification protocol this mode of work required, and documents the failures, which we regard as an essential part of the report.
