Assume I have:
$$H(p_1,\ldots,p_N)=H(q_1,\ldots,q_N)$$
where $H$ is the Shannon Entropy.
Does that mean that I necessarily have the $p_i$ and $q_i$ linked by a permutation? Or is it not true?
For the case with $N=2$ I know it is true but what for a more general case?
This is not true for a more general case.
With $N=3$ consider for example: $$f(x)=H\left(x, \frac{1-x}{2},\frac{1-x}{2}\right)$$ then $f(0)=\ln(2)$, $f(1/3)=\ln(3)$ and $f(1)=0$.
From there for any $x \in (0,1/3)$ there exists $y \in (1/3,1)$ such that $f(y)=f(x)$ and $\{x,1/2(1-x),1/2(1-x)\} \neq \{ y,1/2(1-y),1/2(1-y)\}$.