  We study limitations of polynomials computed by depth two circuits built over
read-once polynomials (ROPs) and depth three syntactically multi-linear
formulas.
  We prove an exponential lower bound for the size of the
$\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over syntactically
multi-linear $\Sigma\Pi\Sigma^{[N^{8/15}]}$ arithmetic circuits computing a
product of variable disjoint linear forms on $N$ variables. We extend the
result to the case of $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over
ROPs of unbounded depth, where the number of variables with $+$ gates as a
parent in an proper sub formula is bounded by $N^{1/2+1/30}$. We show that the
same lower bound holds for the permanent polynomial. Finally we obtain an
exponential lower bound for the sum of ROPs computing a polynomial in ${\sf
VP}$ defined by Raz and Yehudayoff.
  Our results demonstrate a class of formulas of unbounded depth with
exponential size lower bound against the permanent and can be seen as an
exponential improvement over the multilinear formula size lower bounds given by
Raz for a sub-class of multi-linear and non-multi-linear formulas.
  Our proof techniques are built on the one developed by Raz and later extended
by Kumar et. al.\cite{KMS13} and are based on non-trivial analysis of ROPs
under random partitions. Further, our results exhibit strengths and limitations
of the lower bound techniques introduced by Raz\cite{Raz04a}.
