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Bob Proctor sent me an email explaining that the poset $P$ corresponding to the (unshifted, straight) shape $\lambda = (p+(r-1)b, p+(r-2)b, ..., p+b, p)$ has a product formula for its order polynomial, which can be seen via manipulations on the appropriate determinant. Note that this class includes both rectangles ($b=0$), as well as staircases ($p=1$, $b=1$).