Chains in the noncrossing partition lattice

Details Chains in the noncrossing partition lattice Chains in the noncrossing partition lattice

2007-06-19

arxiv.org/abs/0706.2778v2

Mathematics

Combinatorics

Version 2: Several expository changes made, including changes in the abstract, thanks to helpful suggestions from several read

Scientific paper

We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification of finite Coxeter groups) using basic facts about noncrossing partitions. We solve these recursions for each finite Coxeter group in the classification. Among other results, we obtain a simpler proof of a known uniform formula for the number of maximal chains of noncrossing partitions and a new uniform formula for the number of edges in the noncrossing partition lattice. All of our results extend to the m-divisible noncrossing partition lattice.

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