The Catalan matroid.
September 4, 2002
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 nice Tutte polynomial.
We then generalize our construction to obtain a family of matroids,
which we call "shifted matroids". They arose independently and al-
most simultaneously in the work of Klivans, who showed that they
are precisely the matroids whose independence complex is a shifted
A Dyck path of length 2n is a path in the plane from (0, 0) to (2n, 0), with
steps (1, 1) and (1, -1), that never passes below the x-axis. It is a classical
result (see for example [8, Corollary 6.2.3.(iv)]) that the number of Dyck