AI 中文总结
研究关于围栏和圆形围栏序多项式，定义贪婪记录统计量，通过伯恩斯坦基证明相关等式，利用转移得到双射，构造循环记录，最终证明了卡哈内的圆形围栏猜想。
AI 中文摘要
设 \(P_\eps\) 为由路径的一个定向 \(\eps\in\{+,-\}^{n - 1}\) 确定的围栏偏序集。我们在 \(S_n\) 上定义了一个从右到左的贪婪记录统计量 \(\rec_\eps\)，并给出了 \(\sum_{\pi\in S_n}t^{\rec_\eps(\pi)} = n!\Omega(P_\eps;t)\) 的伯恩斯坦基证明。通过转移交织连续阈值过程与端点细化的保序映射，得到明确双射等。还为循环构造了循环记录，证明了卡哈内的圆形围栏猜想。
英文摘要
Let \(P_\eps\) be the fence poset associated with an orientation \(\eps\in\{+,-\}^{n-1}\) of a path. We define a greedy right-to-left record statistic \(\rec_\eps\) on \(S_n\) and prove \[ \sum_{π\in S_n}t^{\rec_\eps(π)}=n!Ω(P_\eps;t), \] by a Bernstein-basis transfer between a continuous threshold recurrence and endpoint-refined order-preserving maps. Under reflection, this statistic agrees pointwise with Kahane's independently obtained greedy block statistic; the alternating specialization gives the zig-zag case posed by Ferroni, Morales, and Panova. A finite form of the transfer yields a direct recursive bijection for \(m\le n\), extended algorithmically to arbitrary alphabets. Refining by record set, direction, and terminal value identifies fixed fibers with decorated endpoint paths and pointed linear extensions of posets whose cover graphs are caterpillars. We also define cyclic records for every nonconstant orientation \(η\) of a cycle and prove \[ \sum_{π\in S_n}t^{\operatorname{crec}_η(π)} =n!Ω(C_η;t). \] When the Hasse diagram is a cycle, reflection identifies these records with Kahane's circular blocks and establishes his circular-fence conjecture.
Comments37 pages
