Noncrossing partition posets

Richard Ehrenborg, Gábor Hetyei

Published 2024-07-11Version 1

We introduce the poset NC^d_n of all noncrossing partitions such that each block has cardinality 1 modulo d and each block of the dual partition also has cardinality 1 modulo d. We obtain the cardinality, the M\"obius function, the rank numbers, the antipode, and the number of maximal chains. Generalizing work of Stanley, we give an edge labeling such that the labels of the maximal chains are exactly the d-parking functions. We also introduce two classes of labeled trees: the first class is in bijective correspondence with the noncrossing partitions in NC^d_n and the second class is in bijective correspondence with the maximal chains.