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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.

# 3. 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.

# 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

# 5. Sage examples

# 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.