- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001439: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 120 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

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

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

$F_{5} = 7\ q^{2} + 10\ q^{3} + 7\ q^{4} + 2\ q^{5}$

$F_{6} = 7\ q^{2} + 22\ q^{3} + 33\ q^{4} + 12\ q^{5} + 2\ q^{6}$

Description

The number of even weak deficiencies and of odd weak exceedences.
For a permutation

$\sigma$ , this is the number of indices

$i$ such that

$\sigma(i) \leq i$ if

$i$ is even and

$\sigma(i) \geq i$ if

$i$ is odd.
According to [1],

$\sigma$ is a D-permutation if all indices have this property and the coefficients of the characteristic polynomial of the homogenized linial arrangement are given by the number of D-permutations with a given number of cycles.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.