2026/07/24 by Pyuyi Chufeng Huang
#math.CO
Let \(P_\eps\) be the fence poset determined by an orientation \(\eps∈\+,-\n-1\) of a path. We define a greedy right-to-left record statistic \(\rec_\eps\) on \(Sn\) and give a Bernstein-basis proof of ∑π∈ Snt\rec_\eps(π)=n!Ω(P_\eps;t), which for alternating signs gives the permutation statistic sought by Ferroni, Morales, and Panova for the zig-zag order polynomial. This generating function was independently obtained by Kahane; after reflection, \(\rec_\eps\) agrees pointwise with his greedy block statistic. Our transfer intertwines a continuous threshold process with endpoint-refined order-preserving maps. Its finite form yields an explicit bijection, and a further refinement by record set, record direction, and terminal value identifies each fixed fiber with decorated endpoint paths and with pointed linear extensions of a record poset. The latter posets are oriented caterpillars; specializing Atkinson's tree-poset recursion determines their terminal spectra, while \(P\)-partition theory gives a quasisymmetric refinement recording inverse descents. Finally, for every nonconstant orientation \(η\) of a cycle we construct cyclic records satisfying ∑π∈ Sntcrecη(π) =n!Ω(Cη;t). For genuine circular fences, reflection identifies these records with the roots of Kahane's circular blocks, proving his circular-fence conjecture.