The Catalan matroid

Abstract

We show how the set of Dyck paths of length 2n naturally gives rise to a matroid, which we call the "Catalan matroid" C_n. We describe this matroid in detail; among several other results, we show that C_n is self-dual, it is representable over the rationals but not over finite fields F_q with q < n-1, and it has a nice Tutte polynomial. We then generalize our construction to obtain a family of matroids, which we call "shifted matroids". They arose independently and almost simultaneously in the work of Klivans, who showed that they are precisely the matroids whose independence complex is a shifted complex.