论文标题
分级图的双重性
Duality of graded graphs through operads
论文作者
论文摘要
成对的渐变图以及分级图对偶的Fomin特性是丰富的组合结构，提供了其他枚举框架。原型示例是整数分区的年轻分级图之一，使我们能够连接数量的标准Young Tableaux和排列数量。在这里，我们使用的是将代数设备提取组成对象组成概念的代数设备来构建成对的分级图。为此，我们首先构造了一对渐变图，其中顶点是语法树，即自由非对称作业的元素。这对图是一个新的二元概念，称为$ ϕ $ -Diagonal对偶性，类似于Fomin引入的偶性。我们还提供了一种一般的方式来从作业中构建成对的分级图，其中潜在的poset类似于年轻的格子。考虑了一些涉及整数构图，Motzkin路径和$ M $树的渐变图的作战示例。
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer partitions, allowing us to connect number of standard Young tableaux and numbers of permutations. Here, we use operads, that algebraic devices abstracting the notion of composition of combinatorial objects, to build pairs of graded graphs. For this, we first construct a pair of graded graphs where vertices are syntax trees, the elements of free nonsymmetric operads. This pair of graphs is dual for a new notion of duality called $ϕ$-diagonal duality, similar to the ones introduced by Fomin. We also provide a general way to build pairs of graded graphs from operads, wherein underlying posets are analogous to the Young lattice. Some examples of operads leading to new pairs of graded graphs involving integer compositions, Motzkin paths, and $m$-trees are considered.
