  The Quantum Symmetric Simple Exclusion Process (QSSEP) is a model of quantum
particles hopping on a finite interval and satisfying the exclusion principle.
Recently Bernard and Jin have studied the fluctuations of the invariant measure
for this process, when the number of sites goes to infinity. These fluctuations
are encoded into polynomials, for which they have given equations and proved
that these equations determine the polynomials completely. In this paper, I
give an explicit combinatorial formula for these polynomials, in terms of
Schr\"oder trees. I also show that, quite surprisingly, these polynomials can
be interpreted as free cumulants of a family of commuting random variables.
