无交叉凸包分拆的秩计数与边点数不变性
Rank Enumeration and Side-Count Invariance for Noncrossing Hull Partitions
研究概述
将凸多边形边界上的有限个点分成若干块，要求不同块的凸包两两不相交。分拆的秩等于点数减去块数。本文计算每个秩上的分拆数量，并证明任意调换各边内部的点数都不改变这些数量。如果每条边都有内部点，各秩的数量还唯一确定各边点数及其重复次数。作为出发点的 Dougherty–Root 多边形构型非秩对称猜想及其互反恒等式，已由 Shivam Patel 证明；本文的新贡献是完整计数及任意边点数置换下的不变性。共线构型不在结论范围内。论文包含完整证明，尚未形式化。
原始公开问题
Dougherty–Root 凸包构型猜想：已知多边形情形解答之后的完整秩计数
Michael Dougherty and Gina Root: Noncrossing Partitions From Hull Configurations — Introduction, unnumbered conjecture after Theorem A; polygonal restriction. The original non-rank-symmetry conclusion is prior work, not claimed as a new solution here.
Shivam Patel: Complete proof for every polygonal hull configuration, with an exact rank-reciprocity formula — August 20, 2026 solution; credited polygonal non-rank-symmetry proof and rank-reciprocity identity.
原文摘要（英文）
Let P be a finite set on the boundary of its convex polygon, with c_i points in the relative interior of side i. We give an explicit formula for the rank polynomial of the partitions of P whose block convex hulls are pairwise disjoint. The formula is a linear evaluation of a product of explicitly defined side polynomials. In particular, it depends only on the multiset of the c_i, not on their cyclic order. Among configurations for which every side has an internal point, equal rank polynomials are equivalent to equal multisets of side counts. The same decomposition gives a direct proof of Patel's rank-reciprocity identity, together with directional rank inequalities and an exact mean-rank formula. The proof groups inclusion-exclusion terms into boundary tilings and separates local forbidden successor arcs from the single cycle that can wind around the entire boundary.
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