  In Catalan percolation, all nearest-neighbor edges $\{i,i+1\}$ along $\mathbb
Z$ are initially occupied, and all other edges are open independently with
probability $p$. Open edges $\{i,j\}$ are occupied if some pair of edges
$\{i,k\}$ and $\{k,j\}$, with $i<k<j$, become occupied. This model was
introduced by Gravner and the third author, in the context of polluted graph
bootstrap percolation.
  We prove that the critical $p_{\mathrm c}$ is strictly between that of
oriented site percolation on $\mathbb Z^2$ and the Catalan growth rate $1/4$.
Our main result shows that an enhanced oriented percolation model, with
non-decaying infinite-range dependency, has a strictly smaller critical
parameter than the classical model. This is reminiscent of the work of
Duminil-Copin, Hil\'ario, Kozma and Sidoravicius on brochette percolation. Our
proof differs, however, in that we do not use Aizenman--Grimmett enhancements
or differential inequalities. Two key ingredients are the work of Hil\'ario,
S\'a, Sanchis and Teixeira on stretched lattices, and the Russo--Seymour--Welsh
result for oriented percolation by Duminil-Copin, Tassion and Teixeira.
