  We consider random matrices whose shape is the dilation $N\lambda$ of a
self-conjugate Young diagram $\lambda$. In the large-$N$ limit, the empirical
distribution of the squared singular values converges almost surely to a
probability distribution $F^{\lambda}$. The moments of $F^{\lambda}$ enumerate
two combinatorial objects: $\lambda$-plane trees and $\lambda$-Dyck paths,
which we introduce and show to be in bijection. We also prove that the
distribution $F^{\lambda}$ is algebraic, in the sense of Rao and Edelman. In
the case of fat hook shapes we provide explicit formulae for $F^{\lambda}$ and
we express it as a free convolution of two measures involving a
Marchenko-Pastur and a Bernoulli distribution.
