It is mentioned in Reverse-Engineering the S-Box of Streebog, Kuznyechik and STRIBOBr1, a mathematical equation to quantify the probability that all coefficients in the DDT of a random 8-bit permutation are at most equal to 8 and that this value occurs at most 25 times as following:
$$P[max()=8 \ and \ (8 \leq25]=\sum_{\ell=0}^{25}\binom{255^2}{\ell}.[ \sum_{d=0}^{3}D(2d)\,]^{255^2 - \ell} \, D(8)^\ell$$ where $$D(d)=\frac{e^{-1/2}}{2^{d/2}(d/2)!} $$
$$ DDT_{i,j} = \#\{x\in\{0,1\}^n | S(x\oplus i)\oplus S(x)=j\}$$
$S(X)$ is referred to Sbox (permutation) in cryptography , $n$ bit size
I understand where D(d) comes from in the paper and i understand why D(2d) is used because DDT coefficients are even (0,2,4,6) but i did not get it how the whole general equation is derived? Is there any reference paper that i can check for deriving the general equation used above? any hint will be helpful