Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000486: Permutations ⟶ ℤ
Values
[[1,2]] => [1,2] => [2,1] => [1,2] => 0
[[1],[2]] => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [3,2,1] => [1,2,3] => 0
[[1,2],[3]] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[[1],[2],[3]] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[[1,3,4],[2]] => [2,1,3,4] => [3,2,4,1] => [1,2,4,3] => 0
[[1,2,4],[3]] => [3,1,2,4] => [4,2,3,1] => [1,2,3,4] => 0
[[1,2,3],[4]] => [4,1,2,3] => [1,3,4,2] => [1,3,4,2] => 1
[[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => [1,4,2,3] => 1
[[1,2],[3,4]] => [3,4,1,2] => [4,1,3,2] => [1,3,2,4] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [4,3,2,1] => [1,2,3,4] => 0
[[1,3],[2],[4]] => [4,2,1,3] => [1,4,3,2] => [1,4,2,3] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 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] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [3,2,4,5,1] => [1,2,4,5,3] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [4,2,3,5,1] => [1,2,3,5,4] => 0
[[1,2,3,5],[4]] => [4,1,2,3,5] => [5,2,3,4,1] => [1,2,3,4,5] => 0
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[[1,3,5],[2,4]] => [2,4,1,3,5] => [3,5,2,4,1] => [1,2,4,3,5] => 0
[[1,2,5],[3,4]] => [3,4,1,2,5] => [4,5,2,3,1] => [1,2,3,4,5] => 0
[[1,3,4],[2,5]] => [2,5,1,3,4] => [3,1,4,5,2] => [1,4,5,2,3] => 0
[[1,2,4],[3,5]] => [3,5,1,2,4] => [4,1,3,5,2] => [1,3,5,2,4] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [5,1,3,4,2] => [1,3,4,2,5] => 1
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [4,3,2,5,1] => [1,2,5,3,4] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => 0
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [5,4,2,3,1] => [1,2,3,4,5] => 0
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,4,3,5,2] => [1,4,2,3,5] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,5,3,4,2] => [1,5,2,3,4] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,3,1,5,2] => [1,5,2,3,4] => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [5,3,1,4,2] => [1,4,2,3,5] => 1
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [5,4,1,3,2] => [1,3,2,4,5] => 0
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,5,4,3,2] => [1,5,2,3,4] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 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] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [3,2,4,5,6,1] => [1,2,4,5,6,3] => 1
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [4,2,3,5,6,1] => [1,2,3,5,6,4] => 1
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [5,2,3,4,6,1] => [1,2,3,4,6,5] => 0
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [6,2,3,4,5,1] => [1,2,3,4,5,6] => 0
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => 1
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [3,5,2,4,6,1] => [1,2,4,6,3,5] => 1
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [4,5,2,3,6,1] => [1,2,3,6,4,5] => 1
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [3,6,2,4,5,1] => [1,2,4,5,3,6] => 1
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [4,6,2,3,5,1] => [1,2,3,5,4,6] => 0
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [5,6,2,3,4,1] => [1,2,3,4,5,6] => 0
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => [1,4,5,6,2,3] => 1
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [4,1,3,5,6,2] => [1,3,5,6,2,4] => 1
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [5,1,3,4,6,2] => [1,3,4,6,2,5] => 1
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [6,1,3,4,5,2] => [1,3,4,5,2,6] => 1
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [4,3,2,5,6,1] => [1,2,5,6,3,4] => 0
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [5,3,2,4,6,1] => [1,2,4,6,3,5] => 1
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [5,4,2,3,6,1] => [1,2,3,6,4,5] => 1
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [6,3,2,4,5,1] => [1,2,4,5,3,6] => 1
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [6,4,2,3,5,1] => [1,2,3,5,4,6] => 0
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [6,5,2,3,4,1] => [1,2,3,4,5,6] => 0
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,4,3,5,6,2] => [1,4,2,3,5,6] => 1
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,5,3,4,6,2] => [1,5,2,3,4,6] => 1
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,6,3,4,5,2] => [1,6,2,3,4,5] => 1
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => 1
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [3,5,1,4,6,2] => [1,4,6,2,3,5] => 1
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [4,5,1,3,6,2] => [1,3,6,2,4,5] => 1
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [3,6,1,4,5,2] => [1,4,5,2,3,6] => 0
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [4,6,1,3,5,2] => [1,3,5,2,4,6] => 1
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [5,6,1,3,4,2] => [1,3,4,2,5,6] => 1
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [4,3,6,2,5,1] => [1,2,5,3,6,4] => 1
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [5,3,6,2,4,1] => [1,2,4,3,6,5] => 0
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [5,4,6,2,3,1] => [1,2,3,4,6,5] => 0
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [6,3,5,2,4,1] => [1,2,4,3,5,6] => 0
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [6,4,5,2,3,1] => [1,2,3,4,5,6] => 0
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [4,3,1,5,6,2] => [1,5,6,2,3,4] => 1
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [5,3,1,4,6,2] => [1,4,6,2,3,5] => 1
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [5,4,1,3,6,2] => [1,3,6,2,4,5] => 1
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [6,3,1,4,5,2] => [1,4,5,2,3,6] => 0
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [6,4,1,3,5,2] => [1,3,5,2,4,6] => 1
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [6,5,1,3,4,2] => [1,3,4,2,5,6] => 1
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,4,6,3,5,2] => [1,4,6,2,3,5] => 1
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,5,6,3,4,2] => [1,5,6,2,3,4] => 1
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,4,2,5,6,3] => [1,4,2,5,6,3] => 1
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,5,2,4,6,3] => [1,5,2,4,6,3] => 1
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,6,2,4,5,3] => [1,6,2,4,5,3] => 1
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [5,4,3,2,6,1] => [1,2,6,3,4,5] => 1
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [6,4,3,2,5,1] => [1,2,5,3,4,6] => 1
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [6,5,3,2,4,1] => [1,2,4,3,5,6] => 0
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [6,5,4,2,3,1] => [1,2,3,4,5,6] => 0
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,5,4,3,6,2] => [1,5,2,3,6,4] => 1
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,6,4,3,5,2] => [1,6,2,3,5,4] => 1
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,6,5,3,4,2] => [1,6,2,3,4,5] => 1
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,2,5,4,6,3] => [1,2,5,3,4,6] => 1
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 1
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => 1
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [1,5,2,6,3,4] => 1
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [5,1,4,2,6,3] => [1,4,2,6,3,5] => 1
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [5,1,6,2,4,3] => [1,6,2,4,3,5] => 1
>>> Load all 273 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 number of cycles of length at least 3 of a permutation.
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
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.
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!