Identifier
-
Mp00083:
Standard tableaux
—shape⟶
Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001559: Permutations ⟶ ℤ
Values
[[1]] => [1] => [[1]] => [1] => 0
[[1,2]] => [2] => [[1,2]] => [1,2] => 0
[[1],[2]] => [1,1] => [[1],[2]] => [2,1] => 0
[[1,2,3]] => [3] => [[1,2,3]] => [1,2,3] => 0
[[1,3],[2]] => [2,1] => [[1,2],[3]] => [3,1,2] => 0
[[1,2],[3]] => [2,1] => [[1,2],[3]] => [3,1,2] => 0
[[1],[2],[3]] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[[1,2,3,4]] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[[1,3,4],[2]] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[[1,2,4],[3]] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[[1,2,3],[4]] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[[1,3],[2,4]] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[[1,2],[3,4]] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[[1,4],[2],[3]] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 0
[[1,3],[2],[4]] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 0
[[1,2],[3],[4]] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 0
[[1],[2],[3],[4]] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[[1,2,3,4,5]] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[[1,2,4,5],[3]] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[[1,2,3,5],[4]] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[[1,2,3,4],[5]] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[[1,3,5],[2,4]] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 1
[[1,2,5],[3,4]] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 1
[[1,3,4],[2,5]] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 1
[[1,2,4],[3,5]] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 1
[[1,2,3],[4,5]] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 1
[[1,4,5],[2],[3]] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 0
[[1,3,5],[2],[4]] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 0
[[1,2,5],[3],[4]] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 0
[[1,3,4],[2],[5]] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 0
[[1,2,4],[3],[5]] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 0
[[1,2,3],[4],[5]] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 0
[[1,4],[2,5],[3]] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[[1,3],[2,5],[4]] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[[1,2],[3,5],[4]] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[[1,3],[2,4],[5]] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[[1,2],[3,4],[5]] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[[1,5],[2],[3],[4]] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 0
[[1,4],[2],[3],[5]] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 0
[[1,3],[2],[4],[5]] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 0
[[1,2],[3],[4],[5]] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 0
[[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[[1,2,3,4,5,6]] => [6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]] => [5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 0
[[1,2,4,5,6],[3]] => [5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 0
[[1,2,3,5,6],[4]] => [5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 0
[[1,2,3,4,6],[5]] => [5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 0
[[1,2,3,4,5],[6]] => [5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 0
[[1,3,5,6],[2,4]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,2,5,6],[3,4]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,3,4,6],[2,5]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,2,4,6],[3,5]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,2,3,6],[4,5]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,3,4,5],[2,6]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,2,4,5],[3,6]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,2,3,5],[4,6]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,2,3,4],[5,6]] => [4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 1
[[1,4,5,6],[2],[3]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,3,5,6],[2],[4]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,2,5,6],[3],[4]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,3,4,6],[2],[5]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,2,4,6],[3],[5]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,2,3,6],[4],[5]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,3,4,5],[2],[6]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,2,4,5],[3],[6]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,2,3,5],[4],[6]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,2,3,4],[5],[6]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 0
[[1,3,5],[2,4,6]] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 3
[[1,2,5],[3,4,6]] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 3
[[1,3,4],[2,5,6]] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 3
[[1,2,4],[3,5,6]] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 3
[[1,2,3],[4,5,6]] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 3
[[1,4,6],[2,5],[3]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,3,6],[2,5],[4]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,2,6],[3,5],[4]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,3,6],[2,4],[5]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,2,6],[3,4],[5]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,4,5],[2,6],[3]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,3,5],[2,6],[4]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,2,5],[3,6],[4]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,3,4],[2,6],[5]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,2,4],[3,6],[5]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,2,3],[4,6],[5]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,3,5],[2,4],[6]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,2,5],[3,4],[6]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,3,4],[2,5],[6]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,2,4],[3,5],[6]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,2,3],[4,5],[6]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 1
[[1,5,6],[2],[3],[4]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,4,6],[2],[3],[5]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,3,6],[2],[4],[5]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,2,6],[3],[4],[5]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,4,5],[2],[3],[6]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,3,5],[2],[4],[6]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,2,5],[3],[4],[6]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,3,4],[2],[5],[6]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,2,4],[3],[5],[6]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,2,3],[4],[5],[6]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 0
[[1,4],[2,5],[3,6]] => [2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2
[[1,3],[2,5],[4,6]] => [2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2
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Description
The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions.
This statistic is the difference between St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. and St000018The number of inversions of a permutation..
A permutation is smooth if and only if this number is zero. Equivalently, this number is zero if and only if the permutation avoids the two patterns $4231$ and $3412$.
This statistic is the difference between St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. and St000018The number of inversions of a permutation..
A permutation is smooth if and only if this number is zero. Equivalently, this number is zero if and only if the permutation avoids the two patterns $4231$ and $3412$.
Map
shape
Description
Sends a tableau to its shape.
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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