- FindStat

edit this map or download as text // json

Identifier

Mp00310: Permutations —toric promotion⟶ Permutations

Images

[1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 13761 entries. <<<

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

Description

Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.

References

[1] Defant, C. Toric Promotion arXiv:2112.06843

Properties

bijective, graded

Sage code

def toggle(pi, i, j):
    if pi.index(i) - pi.index(j) in [-1, 1]:
        return pi
    sigma = [j if e == i else i if e == j else e for e in pi]
    return Permutation(sigma)

def mapping(pi):
    n = len(pi)
    for i in range(1, n):
        pi = toggle(pi, i, i+1)
    return toggle(pi, n, 1)

Weight

Created

Aug 21, 2023 at 10:16 by Martin Rubey

Updated

Aug 21, 2023 at 10:16 by Martin Rubey