Identifier
-
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000246: Permutations ⟶ ℤ (values match St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.)
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [2,1] => 0
[1,1] => [[1],[2]] => [2,1] => [1,2] => 1
[3] => [[1,2,3]] => [1,2,3] => [3,2,1] => 0
[2,1] => [[1,2],[3]] => [3,1,2] => [2,1,3] => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 3
[4] => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 0
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [3,2,1,4] => 3
[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] => 5
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 6
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [4,3,2,1,5] => 4
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,2,1,5,4] => 6
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,2,1,4,5] => 7
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,1,4,3,5] => 8
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,1,3,4,5] => 9
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 10
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 5
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 8
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 9
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 9
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 11
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 12
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 12
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 13
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [2,1,3,4,5,6] => 14
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 15
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 0
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 6
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 10
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 11
[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] => 15
[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] => 18
[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] => 20
[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] => 21
[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] => 0
[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] => 7
[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] => 12
[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] => 13
[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] => 15
[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] => 18
[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] => 16
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => 20
[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] => 22
[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] => 25
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 24
[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] => 27
[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] => 28
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => 0
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => 8
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [9,8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8,9] => 15
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [9,8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8,9] => 21
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [9,8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8,9] => 26
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8,9] => 30
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8,9] => 33
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8,9] => 35
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => 36
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => 0
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [10,1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1,10] => 9
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [9,10,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,10,9] => 16
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [10,9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9,10] => 17
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [10,9,8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8,9,10] => 24
[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [7,8,9,10,1,2,3,4,5,6] => [6,5,4,3,2,1,10,9,8,7] => 24
[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [9,10,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,10,9] => 28
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [10,9,8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8,9,10] => 30
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8,9,10] => 35
[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [9,10,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,10,9] => 32
[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,10,9] => 36
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8,9,10] => 39
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8,9,10] => 42
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,10,9] => 40
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8,9,10] => 44
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 45
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [7,8,9,10,11,12,1,2,3,4,5,6] => [6,5,4,3,2,1,12,11,10,9,8,7] => 36
[6,4,2] => [[1,2,3,4,5,6],[7,8,9,10],[11,12]] => [11,12,7,8,9,10,1,2,3,4,5,6] => [6,5,4,3,2,1,10,9,8,7,12,11] => 44
[4,4,2,2] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]] => [11,12,9,10,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,10,9,12,11] => 52
[4,2,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]] => [11,12,9,10,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,10,9,12,11] => 56
[] => [] => [] => [] => 0
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Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
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.
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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