  We consider models of assignment for random $N$ blue points and $N$ red
points on an interval of length $2N$, in which the cost for connecting a blue
point in $x$ to a red point in $y$ is the concave function $|x-y|^p$, for
$0<p<1$. Contrarily to the convex case $p>1$, where the optimal matching is
trivially determined, here the optimization is non-trivial. The purpose of this
paper is to introduce a special configuration, that we call the \emph{Dyck
matching}, and to study its statistical properties. We compute exactly the
average cost, in the asymptotic limit of large $N$, together with the first
subleading correction. The scaling is remarkable: it is of order $N$ for
$p<\frac{1}{2}$, order $N \ln N$ for $p=\frac{1}{2}$, and $N^{\frac{1}{2}+p}$
for $p>\frac{1}{2}$, and it is universal for a wide class of models. We
conjecture that the average cost of the Dyck matching has the same scaling in
$N$ as the cost of the optimal matching, and we produce numerical data in
support of this conjecture. We hope to produce a proof of this claim in future
work.
