Identifier
-
Mp00258:
Set partitions
—Standard tableau associated to a set partition⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤ
Values
{{1}} => [[1]] => [1] => [1] => 1
{{1,2}} => [[1,2]] => [1,2] => [2,1] => 1
{{1},{2}} => [[1],[2]] => [2,1] => [1,2] => 2
{{1,2,3}} => [[1,2,3]] => [1,2,3] => [3,2,1] => 1
{{1,2},{3}} => [[1,2],[3]] => [3,1,2] => [2,1,3] => 3
{{1,3},{2}} => [[1,3],[2]] => [2,1,3] => [3,1,2] => 2
{{1},{2,3}} => [[1,3],[2]] => [2,1,3] => [3,1,2] => 2
{{1},{2},{3}} => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 3
{{1,2,3,4}} => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 1
{{1,2,3},{4}} => [[1,2,3],[4]] => [4,1,2,3] => [3,2,1,4] => 4
{{1,2,4},{3}} => [[1,2,4],[3]] => [3,1,2,4] => [4,2,1,3] => 3
{{1,2},{3,4}} => [[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => 3
{{1,2},{3},{4}} => [[1,2],[3],[4]] => [4,3,1,2] => [2,1,3,4] => 4
{{1,3,4},{2}} => [[1,3,4],[2]] => [2,1,3,4] => [4,3,1,2] => 2
{{1,3},{2,4}} => [[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => 2
{{1,3},{2},{4}} => [[1,3],[2],[4]] => [4,2,1,3] => [3,1,2,4] => 4
{{1,4},{2,3}} => [[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => 2
{{1},{2,3,4}} => [[1,3,4],[2]] => [2,1,3,4] => [4,3,1,2] => 2
{{1},{2,3},{4}} => [[1,3],[2],[4]] => [4,2,1,3] => [3,1,2,4] => 4
{{1,4},{2},{3}} => [[1,4],[2],[3]] => [3,2,1,4] => [4,1,2,3] => 3
{{1},{2,4},{3}} => [[1,4],[2],[3]] => [3,2,1,4] => [4,1,2,3] => 3
{{1},{2},{3,4}} => [[1,4],[2],[3]] => [3,2,1,4] => [4,1,2,3] => 3
{{1},{2},{3},{4}} => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 4
{{1,2,3,4,5}} => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 1
{{1,2,3,4},{5}} => [[1,2,3,4],[5]] => [5,1,2,3,4] => [4,3,2,1,5] => 5
{{1,2,3,5},{4}} => [[1,2,3,5],[4]] => [4,1,2,3,5] => [5,3,2,1,4] => 4
{{1,2,3},{4,5}} => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,2,1,5,4] => 4
{{1,2,3},{4},{5}} => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,2,1,4,5] => 5
{{1,2,4,5},{3}} => [[1,2,4,5],[3]] => [3,1,2,4,5] => [5,4,2,1,3] => 3
{{1,2,4},{3,5}} => [[1,2,4],[3,5]] => [3,5,1,2,4] => [4,2,1,5,3] => 3
{{1,2,4},{3},{5}} => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [4,2,1,3,5] => 5
{{1,2,5},{3,4}} => [[1,2,5],[3,4]] => [3,4,1,2,5] => [5,2,1,4,3] => 3
{{1,2},{3,4,5}} => [[1,2,5],[3,4]] => [3,4,1,2,5] => [5,2,1,4,3] => 3
{{1,2},{3,4},{5}} => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,1,4,3,5] => 5
{{1,2,5},{3},{4}} => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [5,2,1,3,4] => 4
{{1,2},{3,5},{4}} => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,1,5,3,4] => 4
{{1,2},{3},{4,5}} => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,1,5,3,4] => 4
{{1,2},{3},{4},{5}} => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,1,3,4,5] => 5
{{1,3,4,5},{2}} => [[1,3,4,5],[2]] => [2,1,3,4,5] => [5,4,3,1,2] => 2
{{1,3,4},{2,5}} => [[1,3,4],[2,5]] => [2,5,1,3,4] => [4,3,1,5,2] => 2
{{1,3,4},{2},{5}} => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [4,3,1,2,5] => 5
{{1,3,5},{2,4}} => [[1,3,5],[2,4]] => [2,4,1,3,5] => [5,3,1,4,2] => 2
{{1,3},{2,4,5}} => [[1,3,5],[2,4]] => [2,4,1,3,5] => [5,3,1,4,2] => 2
{{1,3},{2,4},{5}} => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [3,1,4,2,5] => 5
{{1,3,5},{2},{4}} => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [5,3,1,2,4] => 4
{{1,3},{2,5},{4}} => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,1,5,2,4] => 4
{{1,3},{2},{4,5}} => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,1,5,2,4] => 4
{{1,3},{2},{4},{5}} => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [3,1,2,4,5] => 5
{{1,4,5},{2,3}} => [[1,3,5],[2,4]] => [2,4,1,3,5] => [5,3,1,4,2] => 2
{{1,4},{2,3,5}} => [[1,3,5],[2,4]] => [2,4,1,3,5] => [5,3,1,4,2] => 2
{{1,4},{2,3},{5}} => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [3,1,4,2,5] => 5
{{1,5},{2,3,4}} => [[1,3,4],[2,5]] => [2,5,1,3,4] => [4,3,1,5,2] => 2
{{1},{2,3,4,5}} => [[1,3,4,5],[2]] => [2,1,3,4,5] => [5,4,3,1,2] => 2
{{1},{2,3,4},{5}} => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [4,3,1,2,5] => 5
{{1,5},{2,3},{4}} => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,1,5,2,4] => 4
{{1},{2,3,5},{4}} => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [5,3,1,2,4] => 4
{{1},{2,3},{4,5}} => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,1,5,2,4] => 4
{{1},{2,3},{4},{5}} => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [3,1,2,4,5] => 5
{{1,4,5},{2},{3}} => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [5,4,1,2,3] => 3
{{1,4},{2,5},{3}} => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => 3
{{1,4},{2},{3,5}} => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => 3
{{1,4},{2},{3},{5}} => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,1,2,3,5] => 5
{{1,5},{2,4},{3}} => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => 3
{{1},{2,4,5},{3}} => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [5,4,1,2,3] => 3
{{1},{2,4},{3,5}} => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => 3
{{1},{2,4},{3},{5}} => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,1,2,3,5] => 5
{{1,5},{2},{3,4}} => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => 3
{{1},{2,5},{3,4}} => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => 3
{{1},{2},{3,4,5}} => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [5,4,1,2,3] => 3
{{1},{2},{3,4},{5}} => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,1,2,3,5] => 5
{{1,5},{2},{3},{4}} => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,1,2,3,4] => 4
{{1},{2,5},{3},{4}} => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,1,2,3,4] => 4
{{1},{2},{3,5},{4}} => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,1,2,3,4] => 4
{{1},{2},{3},{4,5}} => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,1,2,3,4] => 4
{{1},{2},{3},{4},{5}} => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 5
{{1,2,3,4,5,6}} => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 1
{{1,2,3,4,5},{6}} => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 6
{{1,2,3,4,6},{5}} => [[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [6,4,3,2,1,5] => 5
{{1,2,3,4},{5,6}} => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 5
{{1,2,3,4},{5},{6}} => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 6
{{1,2,3,5,6},{4}} => [[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [6,5,3,2,1,4] => 4
{{1,2,3,5},{4,6}} => [[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [5,3,2,1,6,4] => 4
{{1,2,3,5},{4},{6}} => [[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [5,3,2,1,4,6] => 6
{{1,2,3,6},{4,5}} => [[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [6,3,2,1,5,4] => 4
{{1,2,3},{4,5,6}} => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 4
{{1,2,3},{4,5},{6}} => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 6
{{1,2,3,6},{4},{5}} => [[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [6,3,2,1,4,5] => 5
{{1,2,3},{4,6},{5}} => [[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [3,2,1,6,4,5] => 5
{{1,2,3},{4},{5,6}} => [[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [3,2,1,6,4,5] => 5
{{1,2,3},{4},{5},{6}} => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 6
{{1,2,4,5,6},{3}} => [[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [6,5,4,2,1,3] => 3
{{1,2,4,5},{3,6}} => [[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [5,4,2,1,6,3] => 3
{{1,2,4,5},{3},{6}} => [[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [5,4,2,1,3,6] => 6
{{1,2,4,6},{3,5}} => [[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [6,4,2,1,5,3] => 3
{{1,2,4},{3,5,6}} => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => 3
{{1,2,4},{3,5},{6}} => [[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [4,2,1,5,3,6] => 6
{{1,2,4,6},{3},{5}} => [[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [6,4,2,1,3,5] => 5
{{1,2,4},{3,6},{5}} => [[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [4,2,1,6,3,5] => 5
{{1,2,4},{3},{5,6}} => [[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [4,2,1,6,3,5] => 5
{{1,2,4},{3},{5},{6}} => [[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [4,2,1,3,5,6] => 6
{{1,2,5,6},{3,4}} => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [6,5,2,1,4,3] => 3
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Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
This statistic is undefined for the empty permutation.
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
Standard tableau associated to a set partition
Description
Sends a set partition to the associated standard tableau.
The $j$th column of the standard tableau associated to a set partition is the set of $j$th smallest elements of its blocks arranged in increassing order.
The $j$th column of the standard tableau associated to a set partition is the set of $j$th smallest elements of its blocks arranged in increassing order.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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