  We investigate a class of Young diagrams growing via the addition of unit
cells and satisfying the constraint that the height difference between adjacent
columns $\geq r$. In the long time limit, appropriately re-scaled Young
diagrams approach a limit shape that we compute for each integer $r\geq 0$. We
also determine limit shapes of `diffusively' growing Young diagrams satisfying
the same constraint and evolving through the addition and removal of cells that
proceed with equal rates.
