  In this paper we introduce the definition of marked permutations. We first
present a bijection between Stirling permutations and marked permutations. We
then present an involution on Stirling derangements. Furthermore, we present a
symmetric bivariate enumerative polynomials on $r$-colored marked permutations.
Finally, we give an explanation of $r$-colored marked permutations by using the
language of combinatorial objects.
