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 &#34;Catalan matroid&#34; 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 &#34;shifted matroids&#34;. 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.