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Identifier

Mp00265: Permutations —conjugation by long cycle⟶ Permutations

Images

[1] => [1]

[1,2] => [1,2]

[2,1] => [2,1]

[1,2,3] => [1,2,3]

[1,3,2] => [3,2,1]

[2,1,3] => [1,3,2]

[2,3,1] => [2,3,1]

[3,1,2] => [3,1,2]

[3,2,1] => [2,1,3]

[1,2,3,4] => [1,2,3,4]

[1,2,4,3] => [4,2,3,1]

[1,3,2,4] => [1,2,4,3]

[1,3,4,2] => [3,2,4,1]

[1,4,2,3] => [4,2,1,3]

[1,4,3,2] => [3,2,1,4]

[2,1,3,4] => [1,3,2,4]

[2,1,4,3] => [4,3,2,1]

[2,3,1,4] => [1,3,4,2]

[2,3,4,1] => [2,3,4,1]

[2,4,1,3] => [4,3,1,2]

[2,4,3,1] => [2,3,1,4]

[3,1,2,4] => [1,4,2,3]

[3,1,4,2] => [3,4,2,1]

[3,2,1,4] => [1,4,3,2]

[3,2,4,1] => [2,4,3,1]

[3,4,1,2] => [3,4,1,2]

[3,4,2,1] => [2,4,1,3]

[4,1,2,3] => [4,1,2,3]

[4,1,3,2] => [3,1,2,4]

[4,2,1,3] => [4,1,3,2]

[4,2,3,1] => [2,1,3,4]

[4,3,1,2] => [3,1,4,2]

[4,3,2,1] => [2,1,4,3]

[1,2,3,4,5] => [1,2,3,4,5]

[1,2,3,5,4] => [5,2,3,4,1]

[1,2,4,3,5] => [1,2,3,5,4]

[1,2,4,5,3] => [4,2,3,5,1]

[1,2,5,3,4] => [5,2,3,1,4]

[1,2,5,4,3] => [4,2,3,1,5]

[1,3,2,4,5] => [1,2,4,3,5]

[1,3,2,5,4] => [5,2,4,3,1]

[1,3,4,2,5] => [1,2,4,5,3]

[1,3,4,5,2] => [3,2,4,5,1]

[1,3,5,2,4] => [5,2,4,1,3]

[1,3,5,4,2] => [3,2,4,1,5]

[1,4,2,3,5] => [1,2,5,3,4]

[1,4,2,5,3] => [4,2,5,3,1]

[1,4,3,2,5] => [1,2,5,4,3]

[1,4,3,5,2] => [3,2,5,4,1]

[1,4,5,2,3] => [4,2,5,1,3]

[1,4,5,3,2] => [3,2,5,1,4]

[1,5,2,3,4] => [5,2,1,3,4]

[1,5,2,4,3] => [4,2,1,3,5]

[1,5,3,2,4] => [5,2,1,4,3]

[1,5,3,4,2] => [3,2,1,4,5]

[1,5,4,2,3] => [4,2,1,5,3]

[1,5,4,3,2] => [3,2,1,5,4]

[2,1,3,4,5] => [1,3,2,4,5]

[2,1,3,5,4] => [5,3,2,4,1]

[2,1,4,3,5] => [1,3,2,5,4]

[2,1,4,5,3] => [4,3,2,5,1]

[2,1,5,3,4] => [5,3,2,1,4]

[2,1,5,4,3] => [4,3,2,1,5]

[2,3,1,4,5] => [1,3,4,2,5]

[2,3,1,5,4] => [5,3,4,2,1]

[2,3,4,1,5] => [1,3,4,5,2]

[2,3,4,5,1] => [2,3,4,5,1]

[2,3,5,1,4] => [5,3,4,1,2]

[2,3,5,4,1] => [2,3,4,1,5]

[2,4,1,3,5] => [1,3,5,2,4]

[2,4,1,5,3] => [4,3,5,2,1]

[2,4,3,1,5] => [1,3,5,4,2]

[2,4,3,5,1] => [2,3,5,4,1]

[2,4,5,1,3] => [4,3,5,1,2]

[2,4,5,3,1] => [2,3,5,1,4]

[2,5,1,3,4] => [5,3,1,2,4]

[2,5,1,4,3] => [4,3,1,2,5]

[2,5,3,1,4] => [5,3,1,4,2]

[2,5,3,4,1] => [2,3,1,4,5]

[2,5,4,1,3] => [4,3,1,5,2]

[2,5,4,3,1] => [2,3,1,5,4]

[3,1,2,4,5] => [1,4,2,3,5]

[3,1,2,5,4] => [5,4,2,3,1]

[3,1,4,2,5] => [1,4,2,5,3]

[3,1,4,5,2] => [3,4,2,5,1]

[3,1,5,2,4] => [5,4,2,1,3]

[3,1,5,4,2] => [3,4,2,1,5]

[3,2,1,4,5] => [1,4,3,2,5]

[3,2,1,5,4] => [5,4,3,2,1]

[3,2,4,1,5] => [1,4,3,5,2]

[3,2,4,5,1] => [2,4,3,5,1]

[3,2,5,1,4] => [5,4,3,1,2]

[3,2,5,4,1] => [2,4,3,1,5]

[3,4,1,2,5] => [1,4,5,2,3]

[3,4,1,5,2] => [3,4,5,2,1]

[3,4,2,1,5] => [1,4,5,3,2]

[3,4,2,5,1] => [2,4,5,3,1]

[3,4,5,1,2] => [3,4,5,1,2]

[3,4,5,2,1] => [2,4,5,1,3]

[3,5,1,2,4] => [5,4,1,2,3]

[3,5,1,4,2] => [3,4,1,2,5]

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click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c^{-1}\pi c$ where $c = (1,\ldots,n)$ is the long cycle.
Given the cycle notation of a permutation, this operation adds 1 (mod n) to each element in the cycles.
In [1], the authors prove an occurrence of the cyclic sieving phenomenon for this map.

References

[1] Sagan, B., Shareshian, J., Wachs, M. L. Eulerian quasisymmetric functions and cyclic sieving MathSciNet:2794036 DOI:10.1016/j.aam.2010.01.013

Properties

bijective, graded

Sage code

def mapping(pi):
    if not pi:
        return pi
    n = len(pi)
    long_cycle = Permutation(list(range(2, n+1)) + [1])
    return long_cycle.inverse()*pi*long_cycle

Weight

Created

Nov 30, 2022 at 19:47 by Nadia Lafreniere

Updated

Mar 02, 2023 at 11:33 by Martin Rubey