Stirling permutations, marked permutations and Stirling derangements

Abstract

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.