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Identifier

St000016: Standard tableaux ⟶ ℤ

Values

[[1]] => 0

[[1,2]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5]] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5,6]] => 15

[[1,3,4,5,6],[2]] => 10

[[1,2,4,5,6],[3]] => 10

[[1,2,3,5,6],[4]] => 10

[[1,2,3,4,6],[5]] => 10

[[1,2,3,4,5],[6]] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1115 entries. <<<

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Description

The number of attacking pairs of a standard tableau.
Note that this is actually a statistic on the underlying partition.
A pair of cells $(c, d)$ of a Young diagram (in English notation) is said to be attacking if one of the following conditions holds:
1. $c$ and $d$ lie in the same row with $c$ strictly to the west of $d$.
2. $c$ is in the row immediately to the south of $d$, and $c$ lies strictly east of $d$.

Code

def statistic(x):
    return len(x.shape().attacking_pairs())

Created

Sep 27, 2011 at 20:00 by Chris Berg

Updated

Oct 16, 2015 at 11:08 by Christian Stump