Exceptional and modern intervals of the Tamari lattice
Exceptional and modern intervals of the Tamari lattice
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Tamari 格子的独特和现代间隔
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发表时间：
2018
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通讯作者：
Baptiste Rognerud
中科院分区：
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作者：
Baptiste Rognerud
本文利用Ch{\^a}tel和Pons最近提出的区间-偏集理论来描述Tamari格中一些有趣的区间族。这些族被定义为避免特定配置的区间集。首先，我们考虑我们所谓的异常区间-偏置集，并表明它们对应于在树形操作中作为非交叉树图像获得的区间。我们还证明了异常区间正是由非交叉分区的偏置集中的区间所引起的Tamari格的区间。在第二部分中，我们介绍了现代和无限现代区间偏集的概念。在Chapoton意义上，我们证明了现代区间与Tamari晶格的新区间是相互映射的。我们推导了Tamari格中新区间的一个内在特征。最后，我们考虑无限现代区间的族我们证明了有多少个大小为n的无限现代区间序集就有多少个有n个内顶点的三叉树。
In this article we use the theory of interval-posets recently introduced by Ch{\^a}tel and Pons in order to describe some interesting families of intervals in the Tamari lattices. These families are defined as interval-posets avoiding specific configurations. At first, we consider what we call exceptional interval-posets and show that they correspond to the intervals which are obtained as images of noncrossing trees in the Dendriform operad. We also show that the exceptional intervals are exactly the intervals of the Tamari lattice induced by intervals in the poset of noncrossing partitions. In the second part we introduce the notion of modern and infinitely modern interval-posets. We show that the modern intervals are in bijection with the new intervals of the Tamari lattice in the sense of Chapoton. We deduce an intrinsic characterization of the new intervals in the Tamari lattice. Finally, we consider the family of what we call infinitely modern intervals and we we prove that there are as many infinitely modern interval-posets of size n as there are ternary trees with n inner vertices.