AI 中文总结
研究三浦折纸翻转图度序列，给出统一构造将度数为\(d\)的顶点数表示为对称多项式\(p_d(m,n)\)，其总度数为\(d - 2\)，在特定区域计算并验证相关多项式及界限，还分析了区域外的计数偏差。
AI 中文摘要
折纸折痕图案的翻转图以平面可折叠的山-谷分配为顶点，边连接通过单个面翻转而不同的两个顶点。该图的一个基本不变量是度序列，它统计每个度的顶点数。在\(m×n\)的三浦折纸上，仅对于小度数，该序列才被称为二元多项式，每个计数通过单独的论证获得，其情况分析随度数增长。本文给出了一种统一的构造，对于每个度数\(d\)，将度数为\(d\)的顶点数表示为所有足够大的\(m,n\)的单个对称多项式\(p_d(m,n)\)。在单个度数界限下，该多项式的总度数为\(d - 2\)，对于\(d\geq5\)，它作为\(m^{d - 2}+n^{d - 2}\)的显式倍数增长；当计数分解为独立的行和列因子时，在此证明了该界限，否则仍然开放。区域是\(m,n\geq\max(d - 1,2)\)；通过\(d = 7\)，以封闭形式计算多项式并在每种情况下验证界限。在该区域以下，计数与\(p_d\)的偏差由一个校正项给出，其首项系数到十一次度数为止是\(-4\)乘以一个巴克斯特数。因此，每个\(p_d\)计算恰好允许\(d\)次单一面翻转的三浦折纸的平面可折叠分配。
英文摘要
The flip graph of an origami crease pattern has the locally flat-foldable mountain-valley assignments as vertices, and an edge joins two of them that differ by a single face flip. A basic invariant of this graph is the degree sequence, which counts the vertices of each degree. On the $m\times n$ Miura-ori, this sequence is known to be a bivariate polynomial only for small degrees, each count obtained by a separate argument. This paper gives one uniform construction that expresses, for every degree $d$, the number of degree-$d$ vertices as a single symmetric polynomial in $(m,n)$ for all sufficiently large $m,n$. Its degree in each variable is $d-2$ unconditionally. Subject to a single degree bound, its total degree is $d-2$ as well, with top-degree part an explicit multiple of $m^{d-2}+n^{d-2}$ for $d\ge5$. The bound is proved here when the count splits into independent row and column factors, and open otherwise. The region is $m,n\ge\max(d-1,2)$. The polynomials are given in closed form through $d=10$, unconditional through $d=8$, where the degree bound holds in every case, and conditional on it beyond. Below this region the count departs from the polynomial. One step below, this departure has leading coefficient $-4$ times a Baxter number through $d=11$. Each such polynomial thus counts the Miura-ori's locally flat-foldable assignments admitting exactly $d$ single face flips.
Comments32 pages, 4 figures, 2 tables. Sequel to arXiv:2606.22614. Code: https://github.com/ChakshuGupta13/lab
