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# 1. 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 one-line 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!\$.

# 2. Properties

• Decorated permutations are in bijection with many other objects, such as total subset permutations, Grassmannian necklaces, positroid, Le-diagrams, and bounded affine permutations
• Every decorated permutation can be decomposed into a set of decorated fixed points and a derangement.

• 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.

# 4. References

• [BS20] N. Blitvić and E. Steingrímsson, Permutations, Moments, Measures, arXiv:2001.00280

[FHL20] Shishuo Fu, Guo-Niu Han, Zhicong Lin, k-arrangements, 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

# 6. 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.