Bernstein Transfers and Greedy Records for Fence and Circular-Fence Order Polynomials

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 \[ \sum_{\pi\in S_n}t^{\rec_\eps(\pi)}=n!\Omega(P_\eps;t), \] 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 \(\eta\) of a cycle and prove \[ \sum_{\pi\in S_n}t^{\operatorname{crec}_\eta(\pi)} =n!\Omega(C_\eta;t). \] When the Hasse diagram is a cycle, reflection identifies these records with Kahane's circular blocks and establishes his circular-fence conjecture.