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PreprintCC BY 4.0arXiv

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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Cite

BibTeXRIS
BibTeX
@misc{garber2026counting,
  title = {{Counting Boolean Antichains}},
  author = {Julien Garber and Lina S. Goltermann and Daniel Horiatakis and Dennis König and Tal Gottesman},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27126},
  eprint = {2608.27126},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27126},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Counting Boolean Antichains
AU  - Julien Garber
AU  - Lina S. Goltermann
AU  - Daniel Horiatakis
AU  - Dennis König
AU  - Tal Gottesman
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27126
UR  - https://arxiv.org/abs/2608.27126
AB  - 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.
ER  -