NP$^{\#P}$ = $\exists$PP and other remarks about maximized counting

Abstract: We consider the following decision problem DMAX#SAT, and generalizations thereof: given a quantifier-free propositional formula $F(\mathbf{x},\mathbf{y})$, where $\mathbf{x},\mathbf{y}$ are tuples of variables, and a bound $B$, determine if there is $\vec{x}$ such that $#{\mathbf{y} \mid F(\mathbf{x},\mathbf{y})} \geq B$. This is the decision version of the problem of MAX#SAT: finding $\mathbf{x}$ and $B$ for maximal $B$.