The close connection between factorial moments and factorial correlators as integrals of the same underlying correlation function is explored, leading to extensions of sum rules previously suggested. Cumulants, which were previously found to be the fundamental building blocks for moments, have been analogously defined for the correlators also, revealing the true {ital n}-particle correlations. Decomposing the factorial correlators into cumulants, we find that the largest part of the correlators consists of two-particle correlations for NA22 data. The nonstationarity of the correlation function is found to affect the results to a surprisingly small degree. It is pointed out that all linking schemes for higher-order correlations must be tested not on the correlators but on the cumulants in order to claim success. Finally, applying our scheme to UA1 factorial moment data, we predict the size and shape of UA1 correlators.