  We introduce a class of models defined on ladders with a diagonal structure
generated by $n_p$ plaquettes. The case $n_p=1$ corresponds to the necklace
ladder and has remarkable properties which are studied using DMRG and recurrent
variational ansatzes. The AF Heisenberg model on this ladder is equivalent to
the alternating spin-1/spin-1/2 AFH chain which is known to have a
ferrimagnetic ground state (GS). For doping 1/3 the GS is a fully doped (1,1)
stripe with the holes located mostly along the principal diagonal while the
minor diagonals are occupied by spin singlets. This state can be seen as a Mott
insulator of localized Cooper pairs on the plaquettes. A physical picture of
our results is provided by a $t_p-J_p$ model of plaquettes coupled diagonally
with a hopping parameter $t_d$. In the limit $t_d \to \infty$ we recover the
original $t-J$ model on the necklace ladder while for weak hopping parameter
the model is easily solvable. The GS in the strong hopping regime is
essentially an "on link" Gutzwiller projection of the weak hopping GS. We
generalize the $t_p-J_p-t_d$ model to diagonal ladders with $n_p >1$ and the 2D
square lattice. We use in our construction concepts familiar in Statistical
Mechanics as medial graphs and Bratelli diagrams.
