Sequence of indiscernibles in a theory with an equivalence relation with infinitely many equivalence classes

Question

Let $\mathcal L$ be a language with a single binary relation $E$, and the theory $T$ where $E$ an equivalence relation with infinitely many equivalence classes, each of which is infinite. Are its types of sequences indiscernibles?

Answer 1

Any sequence whose elements are in the same equivalence class is indiscernible. Also any sequence with no two elements in the same equivalence class. (Note these sequences are totally indiscernible i.e., the order does not matter.)

If you ask for sequences of tuples, there are more possibilities, and they are lengthy to describe.