  Work by Gaiotto and Lee, and by Imamura and collaborators suggests that the
superconformal index of $U(N)$ gauge theory should be expressible as a
convergent series whose terms are indices of associated $U(k)$ gauge theories
realized as the worldvolume theories of stacks of $k$ giant-graviton branes. A
different expansion for the index provided by Murthy was shown to hold very
generally, but the connection to the first expansion was not immediately clear.
We study the relation between the two expansions and propose a prescription for
extracting the terms of the first series from those of the second. We follow
this prescription in the case of the $1/2$-BPS index and show that the
contribution of the $m$th term of the first expansion is fully encoded in the
first $m$ terms of the second. In addition, we identify the $m$th term of the
second expansion with the expectation value of the $N$th power of the
superdeterminant in a $U(m|m)$ superunitary matrix integral, which hints at a
brane/anti-brane origin for the term.
