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arXiv · 2407.08577

On the structure of the d-indivisible noncrossing partition posets

Richard Ehrenborg·Gábor Hetyei
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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 and each dual block 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 show that the bijection between the maximal chains and d-parking functions introduced by M\"uhle, Nadeau and Williams gives rise to an EL-labeling. Generalizing a construction of Yan, we also introduce d-parking trees which are in bijective correspondence with the maximal chains.

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Richard Ehrenborg, Gábor Hetyei. 2024-07-11. On the structure of the d-indivisible noncrossing partition posets. https://arxiv.org/abs/2407.08577

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