论文标题
非交叉分区
Noncommutative crossing partitions
论文作者
论文摘要
我们定义和研究非交叉分区，这是非交叉分区的概括。通过在二进制树上引入新的封面关系，我们表明部分有序的一组非交叉分区是一个分级的晶格。这个新的晶格包含kreweras晶格，即非交叉分区的晶格，作为Sublattice。我们通过在晶格上构造明显的$ el $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ - 通过使用$ el $ labeling，我们在Kreweras晶格上恢复了经典结果。我们根据非共同穿越分区的地图来表征Kreweras晶格，Kreweras补体图以及Simion和Ullman定义的两个内态性。我们还建立了三个组合物体之间的关系：标记为$ k+1 $ - 亚洲树，$ k $ chains the lattice和$ k $ -dock tilings。
We define and study noncommutative crossing partitions which are a generalization of non-crossing partitions. By introducing a new cover relation on binary trees, we show that the partially ordered set of noncommutative crossing partitions is a graded lattice. This new lattice contains the Kreweras lattice, the lattice of non-crossing partitions, as a sublattice. We calculate the Möbius function, the number of maximal chains and the number of $k$-chains in this new lattice by constructing an explicit $EL$-labeling on the lattice. By use of the $EL$-labeling, we recover the classical results on the Kreweras lattice. We characterize two endomorphism on the Kreweras lattice, the Kreweras complement map and the involution defined by Simion and Ullman, in terms of the maps on the noncommutative crossing partitions. We also establish relations among three combinatorial objects: labeled $k+1$-ary trees, $k$-chains in the lattice, and $k$-Dyck tilings.
