Identifier
-
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000110: Permutations ⟶ ℤ
Values
[1] => [[1]] => [1] => [1] => 1
[2] => [[1,2]] => [1,2] => [2,1] => 2
[1,1] => [[1],[2]] => [2,1] => [1,2] => 1
[3] => [[1,2,3]] => [1,2,3] => [3,2,1] => 6
[2,1] => [[1,2],[3]] => [3,1,2] => [2,1,3] => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 24
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [3,2,1,4] => 6
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => 4
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [2,1,3,4] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 120
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [4,3,2,1,5] => 24
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,2,1,5,4] => 12
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,2,1,4,5] => 6
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,1,4,3,5] => 4
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,1,3,4,5] => 2
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 720
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 120
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 48
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 24
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 36
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 12
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 6
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 8
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 4
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [2,1,3,4,5,6] => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 5040
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 720
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 240
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 120
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 24
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => 6
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 1
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 40320
[7,1] => [[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => 5040
[6,2] => [[1,2,3,4,5,6],[7,8]] => [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 1440
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 720
[5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => 720
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => 120
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 576
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => 24
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => 6
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => 2
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 1
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Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak 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 $\tau(i) = \sigma(n+1-i)$.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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.
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