Breaking down the reduced Kronecker coefficients

We resolve three interrelated problems on \emph{reduced Kronecker coefficients} $\bar{g}(\alpha ,\beta ,\gamma )$. First, we disprove the \emph{saturation property} which states that $\bar{g}(N\alpha ,N\beta ,N\gamma )>0$ implies $\bar{g}(\alpha ,\beta ,\gamma )>0$ for all $N>1$. Second, we esimate the maximal $\bar{g}(\alpha ,\beta ,\gamma )$, over all $\mathrm{\mid }\alpha \mathrm{\mid }+\mathrm{\mid }\beta \mathrm{\mid }+\mathrm{\mid }\gamma \mathrm{\mid }=n$. Finally, we show that computing $\bar{g}(\lambda ,\mu ,\nu )$ is strongly $\mathrm{\#}P$-hard, i.e. $\mathrm{\#}P$-hard when the input $(\lambda ,\mu ,\nu )$ is in unary.