# Is random shuffling order preserving with respect to the entropy?

Let $$X_1,X_2$$ be discrete random variables such that $$H(X_1) where $$H()$$ is the entropy. We know that for any random mapping $$T$$ which is invertible and independent of a random variable $$X$$ the following is true $$H(T(X))\geq H(T(X)|T)=H(T^{-1}T(X)|T)=H(X),$$ and we have $$H(T(X_1))\geq H(X_1)\text{ and }H(T(X_2))\geq H(X_2).$$ If $$X_1$$, $$X_2$$, $$T(X_1 )$$ and $$T(X_2 )$$ have the same support, is it true that $$H(T(X_1))\leq H(T(X_2))?$$

Randomly mapping the symbols of $$X$$ into other symbols won't change the entropy of the variable (if the mapping is invertible). You are merely changing notation.