  In \cite{Chen:2016fgi} we proposed a differential operator for the evaluation
of the multi-dimensional residues on isolated (zero-dimensional) poles.In this
paper we discuss some new insight on evaluating the (generalized)
Cachazo-He-Yuan (CHY) forms of the scattering amplitudes using this
differential operator. We introduce a tableau representation for the
coefficients appearing in the proposed differential operator. Combining the
tableaux with the polynomial forms of the scattering equations, the evaluation
of the generalized CHY form becomes a simple combinatoric problem. It is thus
possible to obtain the coefficients arising in the differential operator in a
straightforward way. We present the procedure for a complete solution of the
$n$-gon amplitudes at one-loop level in a generalized CHY form. We also apply
our method to fully evaluate the one-loop five-point amplitude in the maximally
supersymmetric Yang-Mills theory; the final result is identical to the one
obtained by Q-Cut.
