Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000541: Permutations ⟶ ℤ (values match St000542The number of left-to-right-minima of a permutation.)
Values
{{1,2}} => [2,1] => [2,1] => 1
{{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,1,2] => 1
{{1,2},{3}} => [2,1,3] => [1,3,2] => 0
{{1,3},{2}} => [3,2,1] => [3,2,1] => 2
{{1},{2,3}} => [1,3,2] => [2,1,3] => 1
{{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [3,4,1,2] => 1
{{1,2,3},{4}} => [2,3,1,4] => [1,3,4,2] => 0
{{1,2,4},{3}} => [2,4,3,1] => [3,1,4,2] => 1
{{1,2},{3,4}} => [2,1,4,3] => [3,2,1,4] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [1,3,2,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [4,3,1,2] => 2
{{1,3},{2,4}} => [3,4,1,2] => [4,1,2,3] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [1,4,3,2] => 0
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => 3
{{1},{2,3,4}} => [1,3,4,2] => [2,4,1,3] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [1,2,4,3] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [3,4,2,1] => 2
{{1},{2,4},{3}} => [1,4,3,2] => [2,1,4,3] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [2,3,1,4] => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [3,4,5,1,2] => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => [1,3,4,5,2] => 0
{{1,2,3,5},{4}} => [2,3,5,4,1] => [3,4,1,5,2] => 1
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,4,2,1,5] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,3,4,2,5] => 0
{{1,2,4,5},{3}} => [2,4,3,5,1] => [3,5,4,1,2] => 1
{{1,2,4},{3,5}} => [2,4,5,1,3] => [3,5,1,2,4] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,3,5,4,2] => 0
{{1,2,5},{3,4}} => [2,5,4,3,1] => [3,1,5,4,2] => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => [3,2,5,1,4] => 2
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,3,2,5,4] => 0
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,1,4,5,2] => 1
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [3,2,1,5,4] => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [3,2,4,1,5] => 2
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,3,2,4,5] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [4,3,5,1,2] => 2
{{1,3,4},{2,5}} => [3,5,4,1,2] => [4,1,5,2,3] => 1
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [1,4,3,5,2] => 0
{{1,3,5},{2,4}} => [3,4,5,2,1] => [4,5,1,3,2] => 1
{{1,3},{2,4,5}} => [3,4,1,5,2] => [4,5,2,1,3] => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,4,5,2,3] => 0
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [4,3,1,5,2] => 2
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [4,1,2,5,3] => 1
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [4,3,2,1,5] => 3
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,4,3,2,5] => 0
{{1,4,5},{2,3}} => [4,3,2,5,1] => [5,4,3,1,2] => 3
{{1,4},{2,3,5}} => [4,3,5,1,2] => [5,4,1,2,3] => 2
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,5,4,3,2] => 0
{{1,5},{2,3,4}} => [5,3,4,2,1] => [4,5,3,2,1] => 3
{{1},{2,3,4,5}} => [1,3,4,5,2] => [2,4,5,1,3] => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,2,4,5,3] => 0
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [4,3,5,2,1] => 3
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [2,4,1,5,3] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [2,4,3,1,5] => 1
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,2,4,3,5] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [5,3,4,1,2] => 2
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [5,1,4,2,3] => 1
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [5,3,1,2,4] => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [1,5,3,4,2] => 0
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,4,3,2,1] => 4
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [2,5,4,1,3] => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [2,5,1,3,4] => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,2,5,4,3] => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [3,5,4,2,1] => 2
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [2,1,5,4,3] => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [2,3,5,1,4] => 1
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,3,5,4] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [3,4,5,2,1] => 2
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [2,1,4,5,3] => 1
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [2,3,1,5,4] => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [2,3,4,1,5] => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [3,4,5,6,1,2] => 1
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [1,3,4,5,6,2] => 0
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [3,4,5,1,6,2] => 1
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [3,4,5,2,1,6] => 2
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [1,3,4,5,2,6] => 0
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [3,4,6,5,1,2] => 1
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [3,4,6,1,2,5] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [1,3,4,6,5,2] => 0
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [3,4,1,6,5,2] => 1
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,4,2,6,1,5] => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [1,3,4,2,6,5] => 0
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [3,4,1,5,6,2] => 1
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,4,2,1,6,5] => 2
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,4,2,5,1,6] => 2
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [1,3,4,2,5,6] => 0
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [3,5,4,6,1,2] => 1
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [3,5,1,6,2,4] => 1
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [1,3,5,4,6,2] => 0
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [3,5,6,1,4,2] => 1
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [3,5,6,2,1,4] => 2
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [1,3,5,6,2,4] => 0
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [3,5,4,1,6,2] => 1
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [3,5,1,2,6,4] => 1
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [3,5,4,2,1,6] => 2
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [1,3,5,4,2,6] => 0
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [3,6,5,4,1,2] => 1
{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [3,6,5,1,2,4] => 1
>>> Load all 411 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 indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Map
Tanimoto
Description
Add 1 to every entry of the permutation (n becomes 1 instead of n+1), except that when n appears at the front or the back of the permutation, instead remove it and place 1 at the other end of the permutation.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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!