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Definition & Example

A decorated permutation of size $n$ is a permutation of $\{1,\dots,n\}$ for which each fixed point is either decorated with a '$+$' or with a '$$'.

We write a decorated permutation in oneline notation as $\tau = [\tau_1,\dots,\tau_n]$ where fixed points $\tau_i = i$ come in two colors '$+$' and '$$'.
the 16 Decorated permutations of size 3  
[+,+,+]  [,+,+]  [+,,+]  [+,+,]  [,,+]  [,+,]  [+,,]  [,,]  
[+,3,2]  [,3,2]  [2,1,+]  [2,1,]  [2,3,1]  [3,1,2]  [3,+,1]  [3,,1] 
 The number of decorated permutations of size $n$ is A000522 and given by $\sum_{k = 0}^n n!/k!\ $.
Properties
 Decorated permutations are in bijection with many other objects, such as total subset permutations, Grassmannian necklaces, positroid, Lediagrams, and bounded affine permutations
 Every decorated permutation can be decomposed into a set of decorated fixed points and a derangement.
Additional information
 In [BS20, FHL20], the authors consider $k$arrangements. These are permutations with fixed points being colored in $k$ colors. In particular, their notion of $2$arrangements coincides with decorated permutations.
References
[BS20] N. Blitvić and E. Steingrímsson, Permutations, Moments, Measures, arXiv:2001.00280
[FHL20] Shishuo Fu, GuoNiu Han, Zhicong Lin, karrangements, statistics and patterns arXiv:2005.06354
[La15] T. Lam, Totally Nonnegative Grassmannian and Grassmannian Polytopes. 1 June 2015. arxiv:1506.00603
[Po06] A. Postnikov, Total positivity, Grassmannians, and networks. 27 Sep 2006. arxiv:0609764
Sage examples
Technical information for database usage
 A decorated permutation is uniquely represented as a list.
 Decorated permutations are graded by their size.
 The database contains all decorated permutations of size at most 6.