The Catalan matroid

  • Authors:

  • Federico Ardila

  • Affiliations:

  • Department of Mathematics, Massacusetts Institute of Technology, Cambridge, MA

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2003

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Abstract

We show how the set of Dyck paths of length 2n naturally gives rise to a matroid, which we call the "Catalan matroid" Cn. We describe this matroid in detail; among several other results, we show that Cn is self-dual, it is representable over Q but not over finite fields Fq with q ≤ n - 2, and it has a remarkably nice Tutte polynomial. We then generalize our construction to obtain a family of matroids, which we call "shifted matroids". They are precisely the matroids whose independence complex is a shifted simplicial complex.