- FindStat

Identifier

Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000462: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5,6]] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 0

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

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

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

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

[[1,2,3,4,5],[6]] => {{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 0

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

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

[[1,3,4,6],[2,5]] => {{1,3,4,6},{2,5}} => [3,5,4,6,2,1] => 7

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

[[1,2,3,6],[4,5]] => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 8

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

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

[[1,2,3,5],[4,6]] => {{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 0

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

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

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

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

[[1,3,4,6],[2],[5]] => {{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => 7

[[1,2,4,6],[3],[5]] => {{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 8

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

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

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

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

[[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => 0

[[1,3,5],[2,4,6]] => {{1,3,5},{2,4,6}} => [3,4,5,6,1,2] => 0

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

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

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

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

[[1,4,6],[2,5],[3]] => {{1,4,6},{2,5},{3}} => [4,5,3,6,2,1] => 8

[[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => 9

[[1,2,6],[3,5],[4]] => {{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => 11

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

[[1,2,6],[3,4],[5]] => {{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => 7

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

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

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

[[1,3,4],[2,6],[5]] => {{1,3,4},{2,6},{5}} => [3,6,4,1,5,2] => 7

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

[[1,2,3],[4,6],[5]] => {{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => 8

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

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

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

[[1,2,4],[3,5],[6]] => {{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => 0

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

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

[[1,4,6],[2],[3],[5]] => {{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => 8

[[1,3,6],[2],[4],[5]] => {{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => 7

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

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

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

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

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

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

[[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => 0

[[1,4],[2,5],[3,6]] => {{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => 0

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

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

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$F_{2} = 2$

$F_{3} = 3 + q^{2}$

$F_{4} = 5 + 3\ q^{2} + 2\ q^{3}$

$F_{5} = 7 + 5\ q^{2} + 6\ q^{3} + 5\ q^{4} + 3\ q^{6}$

$F_{6} = 11 + 9\ q^{2} + 12\ q^{3} + 14\ q^{4} + 8\ q^{5} + 8\ q^{6} + 6\ q^{7} + 6\ q^{8} + q^{9} + q^{11}$

Description

The major index minus the number of excedences of a permutation.
This occurs in the context of Eulerian polynomials [1].

Map

rows

Description

The set partition whose blocks are the rows of the tableau.

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.