- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000004: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + q$

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

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

$F_{6} = 11 + 12\ q + 16\ q^{2} + 16\ q^{3} + 12\ q^{4} + 6\ q^{5} + 3\ q^{6}$

Description

The major index of a permutation.
This is the sum of the positions of its descents,

$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$
Its generating function is

$[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for

$[k]_q = 1 + q + q^2 + \dots q^{k-1}$ .
A statistic equidistributed with the major index is called Mahonian statistic.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.

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.