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arXiv · 2608.27126

Counting Boolean Antichains

Julien Garber·Lina S. Goltermann·Daniel Horiatakis·Dennis König·Tal Gottesman
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Abstract

We say that an antichain in a lattice $L$ is Boolean if it generates a Boolean sublattice in $L$ in a particularly nice way. These purely combinatorial objects play a role in the representation theory of the incidence algebra of the lattice $L$, as indicated by recent results of Rognerud, Yıldırım and of the last author with Klász, Kleinau and Marczinzik. Given these motivations, it is natural to ask: Can we count and construct Boolean antichains in certain lattices? In this paper we give a construction of Boolean antichains in Boolean lattices, partition lattices and Tamari lattices. Furthermore, we share the surprisingly elegant formulas we found for the number of these Boolean antichains.

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Julien Garber, Lina S. Goltermann, Daniel Horiatakis, Dennis König, Tal Gottesman. 2026-08-28. Counting Boolean Antichains. https://arxiv.org/abs/2608.27126

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