Before revision:  Two important parameters of these distributions are the  partition ratio q = \log Z(\beta_{\max)}{Z(\min)} and the counts c_x = |H^{-1}(x)|.
After revision:  Two important parameters of these distributions are the log partition ratio q = \log Z(\beta_{\max)}{Z(\min)} and the counts c_x = |H^{-1}(x)|.