多重拟阵转移多项式的活动展开
An activities expansion of the transition polynomial of a multimatroid
AI总结：
该研究定义多重拟阵斜类的基相关活动度，得到加权转移多项式的活动展开，结合横贯分解、基等价关系与Kochol相容集理论推导更多表达式，并将结果应用于delta-拟阵与拓扑转移多项式，推广了拟阵Tutte多项式的相关结论。
摘要：
多重拟阵的加权转移多项式是Tutte多项式的推广。通过定义多重拟阵中一个斜类相对于基的活动度，我们得到了加权转移多项式的一种活动展开形式。我们还将多重拟阵的所有横贯集分解为若干横贯子集的并，分解中的每一项都具有布尔格结构，且每个横贯所属的项数仅取决于其部分斜类的大小。通过在基上定义等价关系并扩展Kochol的相容集理论，我们得到了多重拟阵转移多项式的更多表达式。我们将多重拟阵的研究结果应用于推导Morse关于delta-拟阵转移多项式的结论，并得到了由可行集确定的delta-拟阵元素子集布尔格的一个划分。最后，我们阐述了多重拟阵如何从嵌入曲面的图中产生，并应用我们的结果得到了拓扑转移多项式的活动展开。本工作推广了拟阵Tutte多项式的相关结果。
英文摘要：
The weighted transition polynomial of a multimatroid is a generalization of the Tutte polynomial. By defining the activity of a skew class with respect to a basis in a multimatroid, we obtain an activities expansion for the weighted transition polynomial. We also decompose the set of all transversals of a multimatroid as a union of subsets of transversals. Each term in the decomposition has the structure of a boolean lattice, and each transversal belongs to a number of terms depending only on the sizes of some of its skew classes. Further expressions for the transition polynomial of a multimatroid are obtained via an equivalence relation on its bases and by extending Kochol's theory of compatible sets. We apply our multimatroid results to obtain a result of Morse about the transition polynomial of a delta-matroid and get a partition of the boolean lattice of subsets of elements of a delta-matroid determined by the feasible sets. Finally, we describe how multimatroids arise from graphs embedded in surfaces and apply our results to obtain an activities expansion for the topological transition polynomial. Our work extends results for the Tutte polynomial of a matroid.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
