The binomial distribution is written as
$p(r|n,\theta )=\binom{n}{r}\theta ^r(1-\theta )^{n-r}$
where $n$ is a positive integer, $0\leq\theta\leq1$, and $r$ is an integer taking values from $0$ to $n$.
I'm trying to find changes of variable that leave this distribution invariant. To illustrate, a general change of variables from $n,r,\theta$ to $m,s,\psi$ would have the form
$\begin{align*}m&=f(n,r,\theta)\\ s&=g(n,r,\theta)\\ \psi&=h(n,r,\theta)\end{align*}$
for some functions $f,g,h$. Here we require that $m,s$ be non-negative integers with $s\leq m$ and that $\psi$ be a real number in the interval $0\leq\psi\leq1$. Moreover the transformation should be a bijection and $h$ should be continuous. Then I say that the binomial distribution is invariant with this change of variables if
$\binom{n}{r}\theta^r(1-\theta)^{n-r}=\binom{m}{s}\psi^s(1-\psi)^{m-s}$
An example would be the following change of variables
$\begin{align*}m&=n\\ s&=n-r\\ \psi&=1-\theta\end{align*}$
Substituting it is easy to verify that the invariance condition holds.
Are there any more such transformations that leave invariant the binomial distribution? Thanks.