AI 中文总结
研究变量分块的块多面体，对路径排列的块在长度和扩张方向进行转移矩阵枚举与埃尔哈特理论分析，给出相关公式，还研究了循环类似物，阐述了路径和循环多面体的一些性质及相关组合解释与开放问题。
AI 中文摘要
我们研究变量被划分为等大小块且局部不等式限制相邻块总贡献的块多面体。对于沿路径排列的块，我们在长度方向上开发了一种转移矩阵枚举方法，并在扩张方向上进行了埃尔哈特理论分析。原始转移矩阵可压缩为加权高度矩阵，长度生成函数的分子和分母由显式递推和行列式公式描述。我们还研究了循环类似物，其长度生成函数由同一行列式的对数导数控制。在埃尔哈特方面，具有至少两个块的路径多面体和偶数循环多面体是完美图的稳定集多面体；因此它们是余度为$2a + 1$的戈伦斯坦多面体（与块数$m \ge 2$无关），满足显式的埃尔哈特 - 麦克唐纳互反性，并且具有次数为$a(m - 2)$的回文单峰$h^*$多项式。奇数循环多面体的分母恰好为二，其格点枚举器是周期整除二的埃尔哈特拟多项式。还讨论了进一步的组合解释和开放问题。
英文摘要
We study block polytopes whose variables are divided into equal-size blocks and whose local inequalities bound the total contribution of adjacent blocks. For blocks arranged along a path, we develop a transfer-matrix enumeration in the length direction. We also carry out an Ehrhart-theoretic analysis in the dilation direction. The original transfer matrix admits a compression to a weighted height matrix, and the numerator and denominator of the length generating function are described by explicit recurrences and determinant formulas. We also study the cyclic analogue, where the length generating function is governed by the logarithmic derivative of the same determinant. On the Ehrhart side, the path polytopes with at least two blocks and the even cyclic polytopes are stable-set polytopes of perfect graphs; consequently they are Gorenstein of codegree $2a + 1$ (independent of the number of blocks $m \ge 2$), satisfy an explicit Ehrhart--Macdonald reciprocity, and have palindromic unimodal $h^*$-polynomials of degree $a(m-2)$. Odd cyclic polytopes have denominator exactly two, and their lattice-point enumerators are Ehrhart quasipolynomials of period dividing two. Further combinatorial interpretations and open problems are discussed.
