  The superconformal index of $\mathcal N=4$ super-Yang Mills theory with
$U(N)$ gauge group can be written as a matrix integral over the gauge group.
Recently, Murthy demonstrated that this integral can be reexpressed as a sum of
terms corresponding to a giant graviton expansion of the index, and provided an
explicit formula for the case of a single giant graviton. Here we give similar
explicit formulae for an arbitrary number, $m\ge1$, of giant gravitons. We
provide 1/2 and 1/16 BPS index examples up to the order where three giant
gravitons contribute and demonstrate that the expansion of the matrix integral
differs from the giant graviton expansion computed in the supergravity dual.
This shows that the giant graviton expansion is not necessarily unique once two
or more giant gravitons start appearing.
