Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000833: Permutations ⟶ ℤ
Values
[[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] => [1,3,2] => 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] => [1,4,2,3] => 2
[[1,2,4],[3]] => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 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] => [1,3,2,4] => 2
[[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] => [1,3,4,2] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2
[[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] => [1,5,2,3,4] => 3
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 2
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 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] => [1,4,2,5,3] => 4
[[1,2,5],[3,4]] => [3,4,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 3
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[[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] => [1,4,5,2,3] => 2
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 4
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 3
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[[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] => [1,3,5,2,4] => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 3
[[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] => [1,3,4,5,2] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 3
[[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] => [1,6,2,3,4,5] => 4
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 2
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,4,6,5] => [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] => [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] => [1,5,2,6,3,4] => 6
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => [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] => [1,5,2,3,6,4] => 5
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,2,4,6,3,5] => [1,2,5,3,6,4] => 4
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 1
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => [1,5,2,3,4,6] => 4
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,2,4,5,3,6] => [1,2,5,3,4,6] => 3
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,3,5,4,6] => [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] => [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] => [1,5,6,2,3,4] => 3
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [1,3,5,6,2,4] => [1,5,2,6,3,4] => 6
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,2,5,6,3,4] => [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] => [1,5,2,3,6,4] => 5
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,2,4,6,3,5] => [1,2,5,3,6,4] => 4
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 1
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,3,4,5,2,6] => [1,5,2,3,4,6] => 4
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,2,4,5,3,6] => [1,2,5,3,4,6] => 3
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,3,5,4,6] => [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] => [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] => [1,4,2,5,3,6] => 6
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,2,5,3,4,6] => [1,2,4,5,3,6] => 2
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => 4
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,2,4,3,5,6] => [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] => [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] => [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] => [1,4,2,6,5,3] => 7
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,2,6,3,5,4] => [1,2,4,6,5,3] => 3
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [1,3,6,2,4,5] => [1,4,2,5,6,3] => 5
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 1
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [1,4,5,2,6,3] => [1,4,6,2,3,5] => 3
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [1,3,5,2,6,4] => [1,4,2,6,3,5] => 6
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,2,5,3,6,4] => [1,2,4,6,3,5] => 2
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [1,3,4,2,6,5] => [1,4,2,3,6,5] => 5
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,2,4,3,6,5] => [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] => [1,2,3,4,6,5] => 1
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,3,5,2,4,6] => [1,4,2,5,3,6] => 6
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,2,5,3,4,6] => [1,2,4,5,3,6] => 2
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => 4
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,2,4,3,5,6] => [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] => [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] => [1,4,5,6,2,3] => 2
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [1,4,6,2,3,5] => [1,4,5,2,6,3] => 4
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [1,3,6,2,4,5] => [1,4,2,5,6,3] => 5
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 1
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,4,5,2,3,6] => [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] => [1,4,2,5,3,6] => 6
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,2,5,3,4,6] => [1,2,4,5,3,6] => 2
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => 4
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,2,4,3,5,6] => [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] => [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] => [1,3,5,2,4,6] => 3
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 6
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 2
>>> Load all 306 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The comajor index of a permutation.
This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
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.
Map
inverse
Description
Sends a permutation to its inverse.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!