Multinomial count of general exact-profile addresses

Exact count of the target addresses at an arbitrary integral profile.

Fix an integral ten-class profile $\beta$ and a scale $m$, and put $N = 3Dm$ with $D=\sum_i n_i \beta_i$. Among the marginal-supported outer addresses of length $N$, those whose $45$-cell joint histogram is exactly the prescribed one are counted by the multinomial coefficient

the product running over the $45$ supported ordered grade triples, with $T_\sigma$ the prescribed multiplicity of $\sigma$.

The point is that an address is determined by the word of joint types it realizes, one supported triple per position, and the prescribed histogram says exactly which words are allowed: the number of such words is the multinomial coefficient of the histogram. The two directions of that correspondence use that the prescribed multiplicity of an unsupported triple is zero, so no information is lost in restricting attention to the $45$ supported cells, and that an exact-profile address is automatically marginally regular.

Together with the factorization $N!/\prod_\sigma T_\sigma! = \bigl(N!/\prod_j M_j!\bigr)\cdot D_*$, where $D_*=\prod_j M_j!/\prod_\sigma T_\sigma!$ is the star degree, this is the numerator of the target-to-ambient ratio that governs how many addresses survive the outer hash. At the fixed witness it specializes to the published count.