We study limitations of polynomials computed by depth two circuits built over

read-once formulas (ROFs). In particular,

1) We prove a 2^{\Omega(n/\log n)} lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff ~[CC, 2008].

2) We obtain a 2^{Omega(sqrt{n})} lower bound on the size of \Sigma\Pi^{[n^{1/15}]} arithmetic circuits built over restricted ROFs of unbounded depth computing the permanent of an nxn matrix (superscripts on gates denote bound on the fan-in). The restriction is on the number of variables with + gates as a parent in a proper sub formula of the ROF to be bounded by \sqrt{n}. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial.

3) We also show an exponential lower bound for the above model against a polynomial in VP.

4) Finally we observe that the techniques developed yield an exponential lower bound on the size of \Sigma\Pi^{[N^{1/30}]} arithmetic circuits built over syntactically multi-linear \Sigma\Pi\Sigma^{[N^{1/4}]} arithmetic circuits computing a product of variable disjoint linear forms on N variables, where the superscripts on gates denote bound on the fan-in.

Our proof techniques are built on the measure developed by Kumar et. al.[2016, TOCT] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz~[STOC, 2004].