  In this paper, we study the computational complexity of the commutative
determinant polynomial computed by a class of set-multilinear circuits which we
call regular set-multilinear circuits. Regular set-multilinear circuits are
commutative circuits with a restriction on the order in which they can compute
polynomials. A regular circuit can be seen as the commutative analogue of the
ordered circuit defined by Hrubes,Wigderson and Yehudayoff [HWY10]. We show
that if the commutative determinant polynomial has small representation in the
sum of constantly many regular set-multilinear circuits, then the commutative
permanent polynomial also has a small arithmetic circuit.
