  In string theories, one-loop scattering amplitudes are characterized by
integrals over genus-one surfaces using the Kronecker-Eisenstein series. A
recent methodology proposed a genus-one basis formed from products of these
series of chain topologies. A prior work further deconstructed cyclic products
of the Kronecker-Eisenstein series on this basis. Building on it, our study
further employs advanced and comprehensive combinatorial techniques to
decompose more general genus-one integrands including a product of an arbitrary
number of cyclic products of Kronecker-Eisenstein series, supplemented by {\tt
Mathematica} codes. Our insights enhance the understanding of multiparticle
amplitudes across various string theories and illuminate loop-level parallels
with string tree-level amplitudes.
