On the structure of the d-indivisible noncrossing partition posets

TL;DR

This paper explores the structure of d-indivisible noncrossing partition posets, providing new enumerative formulas, Hopf algebra antipodes, and combinatorial labelings related to d-parking functions and trees.

Contribution

It introduces a generating function approach for enumeration, computes the antipode in the Hopf algebra, and establishes a bijection with d-parking trees, extending prior combinatorial and algebraic results.

Findings

Derived new formulas for cardinality, M"obius function, and rank numbers.

Computed the antipode of the associated Hopf algebra.

Established a bijection between maximal chains and d-parking trees.

Abstract

We study the poset of d-indivisible noncrossing partitions introduced by M\"uhle, Nadeau and Williams. These are noncrossing partitions such that each block has cardinality 1 modulo d and each block of the dual partition also has cardinality 1 modulo d. Generalizing the work of Speicher, we introduce a generating function approach to reach new enumerative results and recover some known formulas on the cardinality, the M\"obius function and the rank numbers. We compute the antipode of the Hopf algebra of d-indivisible noncrossing partition posets. Generalizing work of Stanley, we give an edge labeling such that the labels of the maximal chains are exactly the d-parking functions. This edge labeling induces an EL-labeling. We also introduce d-parking trees which are in bijective correspondence with the maximal chains.