Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability distributions $\alpha$, $\beta$ and $\gamma$ in $\mathbb{R}^{n+1}$. The set of possible $\alpha$, $\beta$ and $\gamma$ is a polytope in $\mathbb{R}^{3n+3}$. Does anyone know a list of defining inequalities for it?

For those who don't like the probability language: Let $p_{ijk} \in \mathbb{R}_{\geq 0}^H$ with $\sum p_{ijk}=1$. Set $\alpha_i = \sum_{j} p_{ij(n-i-j)}$, $\beta_j = \sum_k p_{(n-j-k)jk}$ and $\gamma_k = \sum_i p_{i(n-i-k)k}$. What are the inequalities defining the possible vectors $\alpha$, $\beta$ and $\gamma$?