Identifier
-
Mp00085:
Standard tableaux
—Schützenberger involution⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000446: Permutations ⟶ ℤ
Values
[[1]] => [[1]] => [1] => [1] => 0
[[1,2]] => [[1,2]] => [1,2] => [1,2] => 0
[[1],[2]] => [[1],[2]] => [2,1] => [2,1] => 1
[[1,2,3]] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [[1,2],[3]] => [3,1,2] => [1,3,2] => 1
[[1,2],[3]] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 2
[[1],[2],[3]] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 3
[[1,2,3,4]] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]] => [[1,2,3],[4]] => [4,1,2,3] => [1,2,4,3] => 1
[[1,2,4],[3]] => [[1,2,4],[3]] => [3,1,2,4] => [1,3,2,4] => 2
[[1,2,3],[4]] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 3
[[1,3],[2,4]] => [[1,3],[2,4]] => [2,4,1,3] => [2,1,4,3] => 4
[[1,2],[3,4]] => [[1,2],[3,4]] => [3,4,1,2] => [1,3,4,2] => 2
[[1,4],[2],[3]] => [[1,2],[3],[4]] => [4,3,1,2] => [1,4,3,2] => 3
[[1,3],[2],[4]] => [[1,3],[2],[4]] => [4,2,1,3] => [2,4,1,3] => 4
[[1,2],[3],[4]] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 5
[[1],[2],[3],[4]] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 6
[[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]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[[1,2,4,5],[3]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,4,3,5] => 2
[[1,2,3,5],[4]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [1,3,2,4,5] => 3
[[1,2,3,4],[5]] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 4
[[1,3,5],[2,4]] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [1,3,2,5,4] => 4
[[1,2,5],[3,4]] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [1,2,4,5,3] => 2
[[1,3,4],[2,5]] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [2,1,3,5,4] => 5
[[1,2,4],[3,5]] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [2,1,4,3,5] => 6
[[1,2,3],[4,5]] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [1,3,4,2,5] => 3
[[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,3,5,2,4] => 4
[[1,2,5],[3],[4]] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,4,3,2,5] => 5
[[1,3,4],[2],[5]] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [2,1,5,3,4] => 5
[[1,2,4],[3],[5]] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,1,3,5] => 6
[[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 7
[[1,4],[2,5],[3]] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [2,1,5,4,3] => 7
[[1,3],[2,5],[4]] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,5,4,2] => 4
[[1,2],[3,5],[4]] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,4,3,5,2] => 5
[[1,3],[2,4],[5]] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,2,1,5,4] => 8
[[1,2],[3,4],[5]] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => 6
[[1,5],[2],[3],[4]] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,5,4,3,2] => 6
[[1,4],[2],[3],[5]] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [2,5,4,1,3] => 7
[[1,3],[2],[4],[5]] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [3,5,2,1,4] => 8
[[1,2],[3],[4],[5]] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 9
[[1],[2],[3],[4],[5]] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[[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]] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[[1,2,4,5,6],[3]] => [[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,5,4,6] => 2
[[1,2,3,5,6],[4]] => [[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,4,3,5,6] => 3
[[1,2,3,4,6],[5]] => [[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,3,2,4,5,6] => 4
[[1,2,3,4,5],[6]] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 5
[[1,3,5,6],[2,4]] => [[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,4,3,6,5] => 4
[[1,2,5,6],[3,4]] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [1,2,3,5,6,4] => 2
[[1,3,4,6],[2,5]] => [[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,3,2,4,6,5] => 5
[[1,2,4,6],[3,5]] => [[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,3,2,5,4,6] => 6
[[1,2,3,6],[4,5]] => [[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,2,4,5,3,6] => 3
[[1,3,4,5],[2,6]] => [[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [2,1,3,4,6,5] => 6
[[1,2,4,5],[3,6]] => [[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [2,1,3,5,4,6] => 7
[[1,2,3,5],[4,6]] => [[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [2,1,4,3,5,6] => 8
[[1,2,3,4],[5,6]] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,3,4,2,5,6] => 4
[[1,4,5,6],[2],[3]] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => 3
[[1,3,5,6],[2],[4]] => [[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,4,6,3,5] => 4
[[1,2,5,6],[3],[4]] => [[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,5,4,3,6] => 5
[[1,3,4,6],[2],[5]] => [[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,3,2,6,4,5] => 5
[[1,2,4,6],[3],[5]] => [[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,3,5,2,4,6] => 6
[[1,2,3,6],[4],[5]] => [[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,4,3,2,5,6] => 7
[[1,3,4,5],[2],[6]] => [[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [2,1,3,6,4,5] => 6
[[1,2,4,5],[3],[6]] => [[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [2,1,5,3,4,6] => 7
[[1,2,3,5],[4],[6]] => [[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [2,4,1,3,5,6] => 8
[[1,2,3,4],[5],[6]] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 9
[[1,3,5],[2,4,6]] => [[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [2,1,4,3,6,5] => 9
[[1,2,5],[3,4,6]] => [[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [2,1,3,5,6,4] => 7
[[1,3,4],[2,5,6]] => [[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,3,4,2,6,5] => 5
[[1,2,4],[3,5,6]] => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,3,2,5,6,4] => 6
[[1,2,3],[4,5,6]] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,4,5,6,3] => 3
[[1,4,6],[2,5],[3]] => [[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,3,2,6,5,4] => 7
[[1,3,6],[2,5],[4]] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,2,4,6,5,3] => 4
[[1,2,6],[3,5],[4]] => [[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,2,5,4,6,3] => 5
[[1,3,6],[2,4],[5]] => [[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,4,3,2,6,5] => 8
[[1,2,6],[3,4],[5]] => [[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,3,5,2,6,4] => 6
[[1,4,5],[2,6],[3]] => [[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [2,1,3,6,5,4] => 8
[[1,3,5],[2,6],[4]] => [[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [2,1,4,6,3,5] => 9
[[1,2,5],[3,6],[4]] => [[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,1,5,4,3,6] => 10
[[1,3,4],[2,6],[5]] => [[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,3,4,6,2,5] => 5
[[1,2,4],[3,6],[5]] => [[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,3,5,4,2,6] => 6
[[1,2,3],[4,6],[5]] => [[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,4,3,5,2,6] => 7
[[1,3,5],[2,4],[6]] => [[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [2,4,1,3,6,5] => 9
[[1,2,5],[3,4],[6]] => [[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,1,5,3,6,4] => 7
[[1,3,4],[2,5],[6]] => [[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [3,2,1,4,6,5] => 10
[[1,2,4],[3,5],[6]] => [[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [3,2,1,5,4,6] => 11
[[1,2,3],[4,5],[6]] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [2,4,1,5,3,6] => 8
[[1,5,6],[2],[3],[4]] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 6
[[1,4,6],[2],[3],[5]] => [[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,3,6,5,2,4] => 7
[[1,3,6],[2],[4],[5]] => [[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,4,6,3,2,5] => 8
[[1,2,6],[3],[4],[5]] => [[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,5,4,3,2,6] => 9
[[1,4,5],[2],[3],[6]] => [[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [2,1,6,5,3,4] => 8
[[1,3,5],[2],[4],[6]] => [[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [2,4,6,1,3,5] => 9
[[1,2,5],[3],[4],[6]] => [[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [2,5,4,1,3,6] => 10
[[1,3,4],[2],[5],[6]] => [[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => 10
[[1,2,4],[3],[5],[6]] => [[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [3,5,2,1,4,6] => 11
[[1,2,3],[4],[5],[6]] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 12
[[1,4],[2,5],[3,6]] => [[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => 12
[[1,3],[2,5],[4,6]] => [[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => 9
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Description
The disorder of a permutation.
Consider a permutation π=[π1,…,πn] and cyclically scanning π from left to right and remove the elements 1 through n on this order one after the other. The disorder of π is defined to be the number of times a position was not removed in this process.
For example, the disorder of [3,5,2,1,4] is 8 since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.
Consider a permutation π=[π1,…,πn] and cyclically scanning π from left to right and remove the elements 1 through n on this order one after the other. The disorder of π is defined to be the number of times a position was not removed in this process.
For example, the disorder of [3,5,2,1,4] is 8 since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection ϕ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word w1w2...wn, compute the image inductively by starting with ϕ(w1)=w1.
At the i-th step, if ϕ(w1w2...wi)=v1v2...vi, define ϕ(w1w2...wiwi+1) by placing wi+1 on the end of the word v1v2...vi and breaking the word up into blocks as follows.
To compute ϕ([1,4,2,5,3]), the sequence of words is
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
The Foata bijection ϕ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word w1w2...wn, compute the image inductively by starting with ϕ(w1)=w1.
At the i-th step, if ϕ(w1w2...wi)=v1v2...vi, define ϕ(w1w2...wiwi+1) by placing wi+1 on the end of the word v1v2...vi and breaking the word up into blocks as follows.
- If wi+1≥vi, place a vertical line to the right of each vk for which wi+1≥vk.
- If wi+1<vi, place a vertical line to the right of each vk for which wi+1<vk.
To compute ϕ([1,4,2,5,3]), the sequence of words is
- 1
- |1|4→14
- |14|2→412
- |4|1|2|5→4125
- |4|125|3→45123.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
Map
Schützenberger involution
Description
Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.
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.
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