- FindStat

Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]] => 6

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

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

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

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

[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]] => 7

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

[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 6

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

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

[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]] => 6

[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]] => 7

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

[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 6

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

[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 6

[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]] => 6

[1,4,5,3,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]] => 7

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

[1,5,2,4,3] => [1,5,2,4,3] => [[1,2,3],[4],[5]] => 7

[1,5,3,2,4] => [1,5,3,2,4] => [[1,2,4],[3],[5]] => 6

[1,5,3,4,2] => [1,5,2,4,3] => [[1,2,3],[4],[5]] => 7

[1,5,4,2,3] => [1,5,4,2,3] => [[1,2,3],[4],[5]] => 7

[1,5,4,3,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]] => 9

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

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

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

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

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

[2,1,5,4,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]] => 8

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

[2,3,1,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 6

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

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

[2,3,5,1,4] => [1,3,5,2,4] => [[1,2,4],[3,5]] => 6

[2,3,5,4,1] => [1,2,5,4,3] => [[1,2,3],[4],[5]] => 7

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

[2,4,1,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 6

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

[2,4,3,5,1] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 6

[2,4,5,1,3] => [1,3,5,2,4] => [[1,2,4],[3,5]] => 6

[2,4,5,3,1] => [1,2,5,4,3] => [[1,2,3],[4],[5]] => 7

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

[2,5,1,4,3] => [2,5,1,4,3] => [[1,3],[2,4],[5]] => 8

[2,5,3,1,4] => [1,5,3,2,4] => [[1,2,4],[3],[5]] => 6

[2,5,3,4,1] => [1,5,2,4,3] => [[1,2,3],[4],[5]] => 7

[2,5,4,1,3] => [2,5,4,1,3] => [[1,3],[2,4],[5]] => 8

[2,5,4,3,1] => [1,5,4,3,2] => [[1,2],[3],[4],[5]] => 9

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

[3,1,2,5,4] => [3,1,2,5,4] => [[1,2,4],[3,5]] => 6

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

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

[3,1,5,2,4] => [3,1,5,2,4] => [[1,2,4],[3,5]] => 6

[3,1,5,4,2] => [2,1,5,4,3] => [[1,3],[2,4],[5]] => 8

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

[3,2,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]] => 7

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

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

[3,2,5,1,4] => [3,2,5,1,4] => [[1,4],[2,5],[3]] => 7

[3,2,5,4,1] => [2,1,5,4,3] => [[1,3],[2,4],[5]] => 8

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

[3,4,1,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 6

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

[3,4,2,5,1] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 6

[3,4,5,1,2] => [1,3,5,2,4] => [[1,2,4],[3,5]] => 6

[3,4,5,2,1] => [1,2,5,4,3] => [[1,2,3],[4],[5]] => 7

[3,5,1,2,4] => [3,5,1,2,4] => [[1,2,4],[3,5]] => 6

[3,5,1,4,2] => [2,5,1,4,3] => [[1,3],[2,4],[5]] => 8

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Description

The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.

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

Robinson-Schensted insertion tableau

Description

Sends a permutation to its Robinson-Schensted insertion tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding insertion tableau.