Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000446: Permutations ⟶ ℤ
Values
[[1]] => [1] => [1] => 0
[[1,2]] => [1,2] => [1,2] => 0
[[1],[2]] => [2,1] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [1,3,2] => 1
[[1,2],[3]] => [3,1,2] => [1,2,3] => 0
[[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[[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] => 2
[[1,2,4],[3]] => [3,1,2,4] => [1,2,4,3] => 1
[[1,2,3],[4]] => [4,1,2,3] => [1,2,3,4] => 0
[[1,3],[2,4]] => [2,4,1,3] => [1,3,2,4] => 2
[[1,2],[3,4]] => [3,4,1,2] => [1,2,3,4] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [1,4,2,3] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [1,3,2,4] => 2
[[1,2],[3],[4]] => [4,3,1,2] => [1,2,3,4] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[[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
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,2,4,5,3] => 2
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,3,5,4] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]] => [2,4,1,3,5] => [1,3,5,2,4] => 4
[[1,2,5],[3,4]] => [3,4,1,2,5] => [1,2,5,3,4] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [1,3,4,2,5] => 3
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,2,4,3,5] => 2
[[1,2,3],[4,5]] => [4,5,1,2,3] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => 2
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => 4
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,2,5,3,4] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,3,4,2,5] => 3
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,2,4,3,5] => 2
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [1,4,2,5,3] => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,3,2,5,4] => 4
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,2,3,5,4] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,3,2,4,5] => 3
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,5,2,3,4] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,4,2,3,5] => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,3,2,4,5] => 3
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[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] => 4
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,2,4,5,6,3] => 3
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,3,5,6,4] => 2
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,4,6,5] => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [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] => 6
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => 2
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [1,3,4,6,2,5] => 5
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,2,4,6,3,5] => 4
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,2,3,6,4,5] => 1
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => 4
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,2,4,5,3,6] => 3
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,3,5,4,6] => 2
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [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] => 3
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [1,3,5,6,2,4] => 6
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,2,5,6,3,4] => 2
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [1,3,4,6,2,5] => 5
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,2,4,6,3,5] => 4
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,3,6,4,5] => 1
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,3,4,5,2,6] => 4
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,2,4,5,3,6] => 3
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,3,5,4,6] => 2
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [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] => 6
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,2,5,3,4,6] => 2
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [1,3,4,2,5,6] => 4
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,2,4,3,5,6] => 3
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [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
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,3,6,2,5,4] => 7
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,2,6,3,5,4] => 3
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [1,3,6,2,4,5] => 5
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,2,6,3,4,5] => 1
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [1,4,5,2,6,3] => 3
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [1,3,5,2,6,4] => 6
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,2,5,3,6,4] => 2
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [1,3,4,2,6,5] => 5
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,2,4,3,6,5] => 4
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,2,3,4,6,5] => 1
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,3,5,2,4,6] => 6
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,2,5,3,4,6] => 2
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,3,4,2,5,6] => 4
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,2,4,3,5,6] => 3
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [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] => 2
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [1,4,6,2,3,5] => 4
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [1,3,6,2,4,5] => 5
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,2,6,3,4,5] => 1
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,4,5,2,3,6] => 3
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,3,5,2,4,6] => 6
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,2,5,3,4,6] => 2
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,3,4,2,5,6] => 4
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,2,4,3,5,6] => 3
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [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] => 3
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [1,3,2,5,4,6] => 6
>>> Load all 332 entries. <<<
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The disorder of a permutation.
Consider a permutation $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The disorder of $\pi$ 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 $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The disorder of $\pi$ 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
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
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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