Let $\Delta^{d-1}_{\leq} = \{y \in \mathbb{R}^d\ | \sum_{i}y_i = 1, \ y_1 \geq y_2 \geq \dots \geq y_d \geq 0\}$. I want to study the analogue of the Dirichlet distribution over this set; that is the distribution with density $p(y) = \prod_{i=1}^d y_i^{a_i-1}$ for some constants $a_i$.
I am interested in obtaining an expression for the normalizing constant (if possible) as well as to try and understand for which values of $\{a_i\}_{i=1}^d$ this function is integrable over $\Delta^{d-1}_{\leq}$. The only case I have managed to figure out is when $a_1=a_2=\dots=a_d = a$ for some $a > 0$. In this case by symmetry we see that $\int_{\Delta^{d-1}_{\leq}}p(y)dy = B(a)/d!$ using properties of the Dirichlet distribution, where $B$ is the multivariate Beta function
In the general non-symmetric case I have been able to rewrite the integral in terms of iterated integrals as follows:
$$\int_{\Delta^{d-1}_{\leq}} p(y)dy = \int_0^{1/d}\int_{y_d}^{\frac{1-y_d}{d-1}}\int_{y_{d-1}}^{\frac{1-y_d-y_{d-1}}{d-2}} \cdots \int_{y_3}^{\frac{1-\sum_{i=3}^d y_i}{2}} \prod_{i=2}^d y_i^{a_i-1}(1-\sum_{j=2}^dy_j)^{a_1-1}dy_2dy_3\cdots dy_d,$$
but do not know how to simplify or evaluate this. Any help of references would be greatly appreciated.
