Let $X_1,X_2,\cdots,X_m$ be all nonnegative integer solutions of the equation $x_1+x_2+\cdots+x_n=d (n,d \in \mathbb{N^*})$.
For example,if $n=2,d=2$,then $m=3$ and $X_1,X_2,X_3=(0,2),(1,1),(2,0)$ or in any other order.
Let $X_{i,k}$ be the k-th element of $X_i$.Hence if $X_1=(0,2)$ then $X_{1,1}=0,X_{1,2}=2$.
Let $v_{i,j}=X_i^{X_j}=\prod_{k=1}^nX_{i,k}^{X_{j,k}}$ (here $0^0=1$),and $V=V(n,d)=(v_{i,j})_{m\times m}$.
Problem :Prove that
(1)$|V|=det(V)>0$;
(2)The largest prime factor of $|V|$ is less than or equal to $d$;
(3)If $d$ is fixed, then $\log|V|$ is a polynomial of $n$,and deg($\log|V|$)$=d-1$.
I already know that $|V(n,1)|=1,|V(1,d)|=d^d,$ and I believe (but without proof) that:
$|V(n,2)|=2^{2n}$
$|V(n,3)|=2^{n(n-1)}\times 3^{1/2n(n+5)}$
$|V(n,4)|=2^{1/6n(5n^2+12n+31)}\times 3^{n(n-1)}$
$|V(n,5)|=2^{1/6(n-1) (n^3+3n^2+14n)}×3^{1/2 (n-1) (n + n^2)}×5^{1/24(n^4+10n^3+35n^2+74n)}$