Description

The Tamari lattice Tn was originally defined on bracketings of a set of n + 1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is

The Tamari lattice Tn was originally defined on bracketings of a set of n + 1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is known, quoting Knuth, “The enumeration of such paths in Tamari lattices remains mysterious.”
The lengths of maximal chains vary over a great range. In this paper, we focus on the chains with maximum length in these lattices. We establish a bijection between the maximum length chains in the Tamari lattice and the set of standard shifted tableaux of staircase shape. We thus derive an explicit formula for the number of maximum length chains, using the Thrall formula for the number of shifted tableaux. We describe the relationship between chains of maximum length in the Tamari lattice and certain maximal chains in weak Bruhat order on the symmetric group, using standard Young tableaux. Additionally, recently, Bergeron and Pr ́eville-Ratelle introduced a generalized Tamari lattice. Some of the results mentioned above carry over to their generalized Tamari lattice.

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Title
  • Chains of Maximum Length in the Tamari Lattice
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Date Created
2014-10-01
Resource Type
  • Text
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    Identifier
    • Digital object identifier: 10.1090/S0002-9939-2014-12069-7
    • Identifier Type
      International standard serial number
      Identifier Value
      1088-6826
    • Identifier Type
      International standard serial number
      Identifier Value
      0002-9939

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    This is a suggested citation. Consult the appropriate style guide for specific citation guidelines.

    Fishel, Susanna, & Nelson, Luke (2014). CHAINS OF MAXIMUM LENGTH IN THE TAMARI LATTICE. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 142(10), 3343-3353. http://dx.doi.org/10.1090/S0002-9939-2014-12069-7

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