发表机构
Michigan Technological University（密歇根理工大学）
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 Young 图中对角台球轨道的计数，将其与中缀链、二进制自行车空间及多米诺骨牌平铺联系起来，给出精确公式、奇偶性限制及渐近下界。
AI 中文摘要
我们研究了整数分拆 $\lambda$ 的 Young 图内的对角台球轨迹。轨迹的斜率为 $\pm 1$，穿过相邻单元格共享的边时直线通过，并在外部边界反射，直至闭合。设 $\sigma(\lambda)$ 为闭合轨道的数目。这推广了 Gerdes 研究的 Chokwe sona 沙画镜像曲线模型，从矩形网格推广到任意 Young 图。设 $G_\lambda$ 为 $\lambda$ 的单元格邻接图，具有其自然的平面嵌入。我们将台球轨道与 $G_\lambda$ 的中缀链的分量等同起来，并推导出 $\sigma(\lambda) = 1 + \dim \mathcal{B}(G_\lambda) = \mathrm{nullity}\\, L(G_\lambda)$ 在 $\mathbb{F}_2$ 上成立，其中 $\mathcal{B}$ 是二进制自行车空间，$L$ 是模 2 拉普拉斯算子。记 $\lambda^\square$ 为删除 $\lambda$ 的第一行和第一列后得到的图，我们进一步证明 $\sigma(\lambda) = 1 + \mathrm{nullity}_{\mathbb{F}_2} A(G_{\lambda^\square})$。对于矩形，这通过 $\mathbb{F}_2$ 上的斐波那契多项式恒等式恢复了 Gerdes 公式 $\sigma(n^m) = \gcd(m,n)$，并且一般情况下给出了刻画：$\sigma(\lambda) = 1$ 当且仅当 $\lambda^\square$ 具有奇数个多米诺骨牌平铺。利用 $\lambda^\square$ 的棋盘二划分，我们将 $\sigma(\lambda) - 1$ 分解为与 Berkovich-Garvan 的 BG-秩相关的颜色不平衡项和偶数秩亏项。这产生了轨道数的奇偶性限制和下界，并表明对于任何固定的 $d$，渐近地所有分拆都有多于 $d$ 个轨道。我们还证明了 $\sigma(\lambda)$ 至多为 $\lambda$ 的 Durfee 长度，确定了阶梯分拆的 $\sigma(n, n-1, \ldots, 1) = \lceil n/2 \rceil$，并表明邻接零度公式与基域无关。
英文摘要
We study diagonal billiard trajectories inside the Young diagram of an integer partition $λ$. A trajectory has slope $\pm 1$, passes straight through sides shared by adjacent cells, and reflects from the exterior boundary until it closes. Let $σ(λ)$ be the number of closed orbits. This extends the mirror-curve model of Chokwe sona sand drawings studied by Gerdes from rectangular grids to arbitrary Young diagrams. Let $G_λ$ be the cell-adjacency graph of $λ$, with its natural planar embedding. We identify the billiard orbits with the components of the medial link of $G_λ$, and deduce that $σ(λ) = 1 + \dim \mathcal{B}(G_λ) = \mathrm{nullity}\, L(G_λ)$ over $\mathbb{F}_2$, where $\mathcal{B}$ is the binary bicycle space and $L$ the mod-2 Laplacian. Writing $λ^\square$ for the diagram obtained by deleting the first row and column of $λ$, we further prove $σ(λ) = 1 + \mathrm{nullity}_{\mathbb{F}_2} A(G_{λ^\square})$. For rectangles this recovers Gerdes' formula $σ(n^m) = \gcd(m,n)$ via identities for Fibonacci polynomials over $\mathbb{F}_2$, and in general it gives the characterization: $σ(λ) = 1$ if and only if $λ^\square$ has an odd number of domino tilings. Using the checkerboard bipartition of $λ^\square$, we decompose $σ(λ) - 1$ into a color-imbalance term, related to the BG-rank of Berkovich-Garvan, and an even rank-deficiency term. This yields parity restrictions and lower bounds for the orbit number, and shows that for any fixed $d$, asymptotically all partitions have more than $d$ orbits. We also prove that $σ(λ)$ is at most the Durfee length of $λ$, determine $σ(n, n-1, \ldots, 1) = \lceil n/2 \rceil$ for staircase partitions, and show that the adjacency-nullity formula is independent of the ground field.
