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A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

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Abstract

In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135 (2005), 77–92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math. 217 (2000), 367–409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.

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Authors and Affiliations

  1. Dipartimento di Sistemi e Informatica, viale Morgagni 65, 50134, Firenze, Italy

    Elena Barcucci, Antonio Bernini & Maddalena Poneti

  2. Dipartimento di Scienze Matematiche ed Informatiche, Pian dei Mantellini, 44, 53100, Siena, Italy

    Luca Ferrari

Authors
  1. Elena Barcucci
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  2. Antonio Bernini
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  3. Luca Ferrari
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  4. Maddalena Poneti
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Correspondence to Luca Ferrari.

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Barcucci, E., Bernini, A., Ferrari, L. et al. A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations. Order 22, 311–328 (2005). https://doi.org/10.1007/s11083-005-9021-x

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  • DOI: https://doi.org/10.1007/s11083-005-9021-x

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