Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000092: Permutations ⟶ ℤ (values match St000353The number of inner valleys of a permutation.)
Values
[[1]] => [1] => [1] => 1
[[1,2]] => [1,2] => [1,2] => 1
[[1],[2]] => [2,1] => [2,1] => 1
[[1,2,3]] => [1,2,3] => [1,2,3] => 1
[[1,3],[2]] => [2,1,3] => [2,3,1] => 1
[[1,2],[3]] => [3,1,2] => [1,3,2] => 1
[[1],[2],[3]] => [3,2,1] => [3,2,1] => 1
[[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 1
[[1,3,4],[2]] => [2,1,3,4] => [2,3,4,1] => 1
[[1,2,4],[3]] => [3,1,2,4] => [1,3,4,2] => 1
[[1,2,3],[4]] => [4,1,2,3] => [1,2,4,3] => 1
[[1,3],[2,4]] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => 2
[[1,4],[2],[3]] => [3,2,1,4] => [3,4,2,1] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [2,4,3,1] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [1,4,3,2] => 1
[[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[[1,3,4,5],[2]] => [2,1,3,4,5] => [2,3,4,5,1] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,3,4,5,2] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,4,5,3] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[[1,3,5],[2,4]] => [2,4,1,3,5] => [2,4,5,1,3] => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,5,1,2] => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [2,3,5,1,4] => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,3,5,2,4] => 2
[[1,2,3],[4,5]] => [4,5,1,2,3] => [1,4,5,2,3] => 2
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,4,5,2,1] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,5,3,1] => 1
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,4,5,3,2] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [2,3,5,4,1] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,3,5,4,2] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => 1
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,5,2,4,1] => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [4,5,2,3,1] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [4,5,1,3,2] => 2
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [2,5,1,4,3] => 2
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [3,5,1,4,2] => 2
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,5,3,2,1] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [3,5,4,2,1] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [2,5,4,3,1] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,5,4,3,2] => 1
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,3,4,5,6,1] => 1
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,3,4,5,6,2] => 1
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,4,5,6,3] => 1
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,5,6,4] => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [2,4,5,6,1,3] => 2
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [3,4,5,6,1,2] => 2
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [2,3,5,6,1,4] => 2
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,3,5,6,2,4] => 2
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,4,5,6,2,3] => 2
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [2,3,4,6,1,5] => 2
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,3,4,6,2,5] => 2
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,4,6,3,5] => 2
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => 2
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,4,5,6,2,1] => 1
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [2,4,5,6,3,1] => 1
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,4,5,6,3,2] => 1
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [2,3,5,6,4,1] => 1
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,3,5,6,4,2] => 1
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,5,6,4,3] => 1
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [2,3,4,6,5,1] => 1
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,3,4,6,5,2] => 1
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,4,6,5,3] => 1
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => 1
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => 2
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [3,4,6,1,2,5] => 2
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => 2
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => 2
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => 2
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [3,5,6,2,4,1] => 2
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [4,5,6,2,3,1] => 2
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [4,5,6,1,3,2] => 2
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,5,6,1,4,3] => 2
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [3,5,6,1,4,2] => 2
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [3,4,6,2,5,1] => 2
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [2,4,6,3,5,1] => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,4,6,3,5,2] => 2
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,5,6,3,4,1] => 2
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,5,6,3,4,2] => 2
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,5,6,2,4,3] => 2
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [2,4,6,1,5,3] => 2
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [3,4,6,1,5,2] => 2
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [2,3,6,1,5,4] => 2
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,3,6,2,5,4] => 2
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,4,6,2,5,3] => 2
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,5,6,3,2,1] => 1
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [3,5,6,4,2,1] => 1
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [2,5,6,4,3,1] => 1
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,5,6,4,3,2] => 1
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [3,4,6,5,2,1] => 1
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [2,4,6,5,3,1] => 1
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,4,6,5,3,2] => 1
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [2,3,6,5,4,1] => 1
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,3,6,5,4,2] => 1
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 1
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [3,6,2,5,1,4] => 3
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [4,6,2,5,1,3] => 3
>>> Load all 119 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 outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
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
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
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!