论文标题
线性$ k $ - 总和图
Linear $k$-Chord Diagrams
论文作者
论文摘要
我们将线性和弦图的概念概括为匹配的尺寸$ k $的情况，我们称之为$ k $ - 总和图。我们提供正式的生成功能和复发关系，列举了这些$ k $的和弦图，该图的数量是短和弦的数量，其中后者被定义为匹配集的所有成员都相邻，并且是线性和弦图中短和弦或环的概括。我们还按照简短和弦构建的连接组件的数量来列举$ k $ -schord图，并在这种情况下提供相关的生成功能。我们表明，短和弦和连接组件的分布是渐近的泊松，并提供了相关的手段。最后，我们提供了复发关系，列举了不交叉的$ k $ chord图表，该图的数量是简短的和弦的数量，概括了Narayana数字并建立了渐近正态性，提供了相关的均值和差异。还讨论了通用游戏的应用程序。
We generalize the notion of linear chord diagrams to the case of matched sets of size $k$, which we call $k$-chord diagrams. We provide formal generating functions and recurrence relations enumerating these $k$-chord diagrams by the number of short chords, where the latter is defined as all members of the matched set being adjacent, and is the generalization of a short chord or loop in a linear chord diagram. We also enumerate $k$-chord diagrams by the number of connected components built from short chords and provide the associated generating functions in this case. We show that the distributions of short chords and connected components are asymptotically Poisson, and provide the associated means. Finally, we provide recurrence relations enumerating non-crossing $k$-chord diagrams by the number of short chords, generalising the Narayana numbers, and establish asymptotic normality, providing the associated means and variances. Applications to generalized games of memory are also discussed.
