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Identifier

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0

[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0

[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0

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

[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 0

[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0

[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0

[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0

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

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

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

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

[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 0

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

[1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0

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

[1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => [1,4,5,3,2] => 0

[1,4,5,3,2] => [1,5,3,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 0

[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => 0

[1,5,2,4,3] => [1,4,5,2,3] => [1,3,5,2,4] => [1,4,3,5,2] => 0

[1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,4,3,5,2] => 0

[1,5,3,4,2] => [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0

[1,5,4,2,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0

[1,5,4,3,2] => [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0

[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0

[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0

[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0

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

[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,4,5,3] => 0

[2,1,5,4,3] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 0

[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 0

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

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

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

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

[3,2,1,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0

[3,2,1,5,4] => [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0

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

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

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

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

[3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => [3,4,2,1,5] => 0

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

[3,4,2,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 0

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

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

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

[3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => [3,4,2,5,1] => 0

[3,5,1,4,2] => [4,5,1,3,2] => [2,5,1,4,3] => [4,2,5,3,1] => 0

[3,5,2,1,4] => [5,2,3,1,4] => [5,1,3,2,4] => [3,4,2,5,1] => 0

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Description

The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.

Map

first fundamental transformation

Description

Return the permutation whose cycles are the subsequences between successive left to right maxima.

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

Clarke-Steingrimsson-Zeng inverse

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].