Rewriting partitions using exponents

DisplayForm@RowBox@
  Riffle[If[#2 == 1, #1, #1^#2] & @@@
    Tally[HoldForm /@ {2, 2, 1, 50, 50, 50}],
   "\[ThinSpace]"]

$2^2\, 1\, 50^3$

You can also apply TraditionalForm and TeXForm gracefully on this (but only after DisplayForm.

Update

Fix issue pointed out by @ZeroTheHero

Tally[{2, 2, 1, 1}] /. {x_Integer, y_Integer} :> If[y == 1, y, Defer[x^y]]
(* {2^2, 1^2} *)

---

Tally[{2, 2, 1, 1}] /. {x_Integer, y_Integer} :> If[x == 1, x, Defer[x^y]]

(* {2^2, 1} *)

This may be useful:

Times @@ HoldForm /@ {2, 2, 1, 50, 50, 50}

1 2² 50³