  The CHY-integrand of bi-adjoint cubic scalar theory is a product of two
PT-factors. This pair of PT-factors can be interpreted as defining a
permutation. We introduce the cycle representation of permutation in this paper
for the understanding of cubic scalar amplitude. We show that, given a
permutation related to the pair of PT-factors, the pole and vertex information
of Feynman diagrams of corresponding CHY-integrand is completely characterized
by the cycle representation of permutation. Inversely, we also show that, given
a set of Feynman diagrams, the cycle representation of corresponding PT-factor
can be recursively constructed. In this sense, there exists a deep connection
between cycles of a permutation and amplitude. Based on these results, we have
investigated the relations among different independent pairs of PT-factors in
the context of cycle representation as well as the multiplication of
cross-ratio factors.
