Closed formula for the ZRP normalization constant

Derive a closed formula for the normalization constant C_{L,N} of the steady-state distribution P_s(C)=C_{L,N}^{-1}(N!/(A_1!A_2!\cdots A_L!))^2\delta_{N,A} for the zero-range process with D_A=Q_A, beyond the explicitly solvable case L=2.

Background

The paper analyzes a single-species zero-range process whose steady-state distribution factorizes over lattice sites. For the parameter choice D_A=Q_A, the weight of a configuration is proportional to the square of the multinomial factor, and the normalization constant C_{L,N} is required to turn these weights into a probability distribution.

The authors state that a closed formula is known only in the two-site case. Determining C_{L,N} for general system size would provide an exact characterization of this steady state and could facilitate analytic calculations of its observables.