Henri Mühle - Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

dmtcs:6363 - Discrete Mathematics & Theoretical Computer Science, April 22, 2020, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) - https://doi.org/10.46298/dmtcs.6363
Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition LatticesArticle

Authors: Henri Mühle 1

  • 1 Laboratoire d'informatique de l'École polytechnique [Palaiseau]

We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.


Volume: DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
Published on: April 22, 2020
Imported on: July 4, 2016
Keywords: Combinatorics,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
Funding:
    Source : OpenAIRE Graph
  • Fondation Sciences Mathématiques de Paris; Funder: French National Research Agency (ANR); Code: ANR-10-LABX-0098

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