- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000672: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => 0

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

[2,1] => [2,1] => [2,1] => 0

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

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

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

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

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

[3,2,1] => [3,2,1] => [3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].