Combinatorial Statistics on Non-crossing Partitions
Combinatorial Statistics on Non-crossing Partitions
复制标题
非交叉分区的组合统计
DOI：
10.1016/0097-3165(94)90066-3
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
R. Simion
中科院分区：
文献类型：
--
作者：
R. Simion
四个统计量，ls，rb，rs，andlb，以前研究的所有分区的{1，2，.，n}，适用于非交叉分区。我们考虑这些统计量的单一和联合分布，并证明等分布的结果。我们得到的q-和p，q-类似的Catalan和Narayana数，完善的秩对称性和单峰的非交叉分区的晶格。两个单峰的结构，其中之一属于杨氏晶格，陈述。我们展示非交叉分区的统计和建立的置换统计应用于限制排列之间的关系。我们所有的证明都是组合的，依赖于双射对应的构造和非交叉分区格的结构性质。
Four statistics,ls, rb, rs, andlb, previously studied on all partitions of {1, 2, …,n}, are applied to non-crossing partitions. We consider single and joint distributions of these statistics and prove equidistribution results. We obtainq- andp,q-analogues of Catalan and Narayana numbers which refine the rank symmetry and unimodality of the lattice of non-crossing partitions. Two unimodality conjectures, one of which pertains to Young's lattice, are stated. We exhibit relations between statistics on non-crossing partitions and established permutation statistics applied to restricted permutations. All our proofs are combinatorial, relying on the construction of bijective correspondences and on structural properties of the lattice of non-crossing partitions.