Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000004: Permutations ⟶ ℤ
Values
[[1]] => [1] => [1] => [1] => 0
[[1,2]] => [1,2] => [2,1] => [2,1] => 1
[[1],[2]] => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [3,2,1] => [3,2,1] => 3
[[1,3],[2]] => [2,1,3] => [2,3,1] => [3,1,2] => 1
[[1,2],[3]] => [3,1,2] => [1,3,2] => [2,3,1] => 2
[[1],[2],[3]] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 6
[[1,3,4],[2]] => [2,1,3,4] => [3,4,2,1] => [4,3,1,2] => 3
[[1,2,4],[3]] => [3,1,2,4] => [2,4,3,1] => [4,2,3,1] => 4
[[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => [3,4,2,1] => 5
[[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => [3,2,4,1] => 4
[[1,4],[2],[3]] => [3,2,1,4] => [2,3,4,1] => [4,1,2,3] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [1,3,4,2] => [2,4,1,3] => 2
[[1,2],[3],[4]] => [4,3,1,2] => [1,2,4,3] => [2,3,4,1] => 3
[[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] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[[1,3,4,5],[2]] => [2,1,3,4,5] => [4,5,3,2,1] => [5,4,3,1,2] => 6
[[1,2,4,5],[3]] => [3,1,2,4,5] => [3,5,4,2,1] => [5,4,2,3,1] => 7
[[1,2,3,5],[4]] => [4,1,2,3,5] => [2,5,4,3,1] => [5,3,4,2,1] => 8
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => [4,5,3,2,1] => 9
[[1,3,5],[2,4]] => [2,4,1,3,5] => [4,2,5,3,1] => [5,3,4,1,2] => 4
[[1,2,5],[3,4]] => [3,4,1,2,5] => [3,2,5,4,1] => [5,3,2,4,1] => 7
[[1,3,4],[2,5]] => [2,5,1,3,4] => [4,1,5,3,2] => [4,5,3,1,2] => 5
[[1,2,4],[3,5]] => [3,5,1,2,4] => [3,1,5,4,2] => [4,5,2,3,1] => 6
[[1,2,3],[4,5]] => [4,5,1,2,3] => [2,1,5,4,3] => [4,3,5,2,1] => 8
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,4,5,2,1] => [5,4,1,2,3] => 3
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,5,3,1] => [5,2,4,1,3] => 4
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [2,3,5,4,1] => [5,2,3,4,1] => 5
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,4,5,3,2] => [3,5,4,1,2] => 5
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,3,5,4,2] => [3,5,2,4,1] => 6
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => [3,4,5,2,1] => 7
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,4,1,5,2] => [4,5,1,2,3] => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => [4,2,5,1,3] => 4
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,3,1,5,4] => [4,2,3,5,1] => 5
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,4,2,5,3] => [3,4,5,1,2] => 3
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,2,5,4] => [3,4,2,5,1] => 6
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,3,4,5,2] => [2,5,1,3,4] => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,2,4,5,3] => [2,3,5,1,4] => 3
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,5,4] => [2,3,4,5,1] => 4
[[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] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 15
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [5,6,4,3,2,1] => [6,5,4,3,1,2] => 10
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [4,6,5,3,2,1] => [6,5,4,2,3,1] => 11
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [3,6,5,4,2,1] => [6,5,3,4,2,1] => 12
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [2,6,5,4,3,1] => [6,4,5,3,2,1] => 13
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [5,6,4,3,2,1] => 14
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [5,3,6,4,2,1] => [6,5,3,4,1,2] => 7
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [4,3,6,5,2,1] => [6,5,3,2,4,1] => 11
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [5,2,6,4,3,1] => [6,4,5,3,1,2] => 8
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [4,2,6,5,3,1] => [6,4,5,2,3,1] => 9
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [3,2,6,5,4,1] => [6,4,3,5,2,1] => 12
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => [5,6,4,3,1,2] => 9
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [4,1,6,5,3,2] => [5,6,4,2,3,1] => 10
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [3,1,6,5,4,2] => [5,6,3,4,2,1] => 11
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => [5,4,6,3,2,1] => 13
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 6
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [3,5,6,4,2,1] => [6,5,2,4,1,3] => 7
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [3,4,6,5,2,1] => [6,5,2,3,4,1] => 8
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [2,5,6,4,3,1] => [6,3,5,4,1,2] => 8
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [2,4,6,5,3,1] => [6,3,5,2,4,1] => 9
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [2,3,6,5,4,1] => [6,3,4,5,2,1] => 10
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,5,6,4,3,2] => [4,6,5,3,1,2] => 9
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,4,6,5,3,2] => [4,6,5,2,3,1] => 10
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,3,6,5,4,2] => [4,6,3,5,2,1] => 11
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,6,5,4,3] => [4,5,6,3,2,1] => 12
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => 6
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => [5,6,3,2,4,1] => 10
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => [5,4,6,3,1,2] => 8
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => [5,4,6,2,3,1] => 9
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [5,4,3,6,2,1] => 12
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [4,5,2,6,3,1] => [6,4,5,1,2,3] => 4
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [3,5,2,6,4,1] => [6,4,2,5,1,3] => 7
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [3,4,2,6,5,1] => [6,4,2,3,5,1] => 8
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,5,3,6,4,1] => [6,3,4,5,1,2] => 5
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [2,4,3,6,5,1] => [6,3,4,2,5,1] => 9
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [4,5,1,6,3,2] => [5,6,4,1,2,3] => 5
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [3,5,1,6,4,2] => [5,6,2,4,1,3] => 6
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [3,4,1,6,5,2] => [5,6,2,3,4,1] => 7
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,5,1,6,4,3] => [5,3,6,4,1,2] => 8
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [2,4,1,6,5,3] => [5,3,6,2,4,1] => 9
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [2,3,1,6,5,4] => [5,3,4,6,2,1] => 10
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,5,3,6,4,2] => [4,6,3,5,1,2] => 6
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,4,3,6,5,2] => [4,6,3,2,5,1] => 10
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,5,2,6,4,3] => [4,5,6,3,1,2] => 7
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,4,2,6,5,3] => [4,5,6,2,3,1] => 8
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,3,2,6,5,4] => [4,5,3,6,2,1] => 11
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [3,4,5,6,2,1] => [6,5,1,2,3,4] => 3
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [2,4,5,6,3,1] => [6,2,5,1,3,4] => 4
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [2,3,5,6,4,1] => [6,2,3,5,1,4] => 5
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [2,3,4,6,5,1] => [6,2,3,4,5,1] => 6
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,4,5,6,3,2] => [3,6,5,1,2,4] => 5
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,3,5,6,4,2] => [3,6,2,5,1,4] => 6
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,3,4,6,5,2] => [3,6,2,4,5,1] => 7
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,2,5,6,4,3] => [3,4,6,5,1,2] => 7
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,2,4,6,5,3] => [3,4,6,2,5,1] => 8
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,6,5,4] => [3,4,5,6,2,1] => 9
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [4,5,6,1,2,3] => 3
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [3,1,5,2,6,4] => [4,5,2,6,1,3] => 6
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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.
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
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
This map sends the major index to the number of inversions.
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.
searching the database
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