  The zeta and Moebius transforms over the subset lattice of $n$ elements and
the so-called subset convolution are examples of unary and binary operations on
set functions. While their direct computation requires $O(3^n)$ arithmetic
operations, less naive algorithms only use $2^n \mathrm{poly}(n)$ operations,
nearly linear in the input size. Here, we investigate a related $n$-ary
operation that takes $n$ set functions as input and maps them to a new set
function. This operation, we call multi-subset transform, is the core
ingredient in the known inclusion--exclusion recurrence for weighted sums over
acyclic digraphs, which extends Robinson's recurrence for the number of
labelled acyclic digraphs. Prior to this work the best known complexity bound
was the direct $O(3^n)$. By reducing the task to multiple instances of
rectangular matrix multiplication, we improve the complexity to $O(2.985^n)$.
