Queries for Decorated permutations: search statistic / browse statistics / browse maps from / browse maps to

# 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!\$.

# 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.

# 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, 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

# 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.

If you want to edit this wiki page, you can download the raw markdown and send your new version to info@findstat.org