论文标题
在对称下的花束和偶极嵌入的枚举公式的目录
A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings Under Symmetries
论文作者
论文摘要
由DNA折纸引起的问题激发，我们为在不同类型的对称性下的花束和偶极子的各种细胞嵌入的各种细胞嵌入提供了一般计数框架和枚举公式。我们的代数框架可以建设性地用于生成所需的对称类别，我们将Burnside的引理与各种对称组一起得出枚举公式。我们的结果将几个现有公式吸收到了这个统一的框架中。此外，我们为带有彩色边缘的花束提供新的公式（因此是不可取向表面中的花束）以及定向花束的定向嵌入。我们还列举了顶点标记的偶极子嵌入。由于偶极子嵌入可以用排列表示，因此该公式还适用于某些等效类别的排列和排列矩阵。除了与DNA折纸（包括RNA二级结构，Feynman图和拓扑图理论）之外，所产生的花束和偶极对称公式列举了与各种区域相关的结构。对于未颜色的对象，我们对58个不同的序列进行编目，据我们所知，其中43个序列尚未得到描述。
Motivated by a problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside's Lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously.
