Identifier
-
Mp00202:
Integer partitions
—first row removal⟶
Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000872: Permutations ⟶ ℤ
Values
[1,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[2,2] => [2] => [[1,2]] => [1,2] => 0
[2,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[3,2] => [2] => [[1,2]] => [1,2] => 0
[3,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[2,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[2,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[1,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[4,2] => [2] => [[1,2]] => [1,2] => 0
[4,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[3,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[3,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[3,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[2,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[2,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[2,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[5,2] => [2] => [[1,2]] => [1,2] => 0
[5,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[4,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[4,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[4,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[3,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[3,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[3,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[3,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[2,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[2,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[2,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 0
[6,2] => [2] => [[1,2]] => [1,2] => 0
[6,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[5,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[5,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[5,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[4,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[4,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[4,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[4,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[4,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[3,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[3,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[3,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[3,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[3,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[2,2,2,2] => [2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2
[2,2,2,1,1] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 1
[2,2,1,1,1,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 0
[2,1,1,1,1,1,1] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 0
[7,2] => [2] => [[1,2]] => [1,2] => 0
[7,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[6,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[6,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[6,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[5,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[5,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[5,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[5,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[5,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[4,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[4,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[4,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[4,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[4,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[4,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[3,3,3] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 1
[3,3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 1
[3,3,1,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 0
[3,2,2,2] => [2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2
[3,2,2,1,1] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 1
[3,2,1,1,1,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 0
[3,1,1,1,1,1,1] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 0
[8,2] => [2] => [[1,2]] => [1,2] => 0
[8,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[7,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[7,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[7,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[6,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[6,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[6,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[6,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[6,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[5,5] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[5,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[5,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[5,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[5,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[5,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[5,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[4,4,2] => [4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => 1
[4,4,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 0
[4,3,3] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 1
[4,3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 1
[4,3,1,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 0
[4,2,2,2] => [2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2
[4,2,2,1,1] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 1
[4,2,1,1,1,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 0
[4,1,1,1,1,1,1] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 0
[9,2] => [2] => [[1,2]] => [1,2] => 0
[9,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
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Description
The number of very big descents of a permutation.
A very big descent of a permutation $\pi$ is an index $i$ such that $\pi_i - \pi_{i+1} > 2$.
For the number of descents, see St000021The number of descents of a permutation. and for the number of big descents, see St000647The number of big descents of a permutation.. General $r$-descents were for example be studied in [1, Section 2].
A very big descent of a permutation $\pi$ is an index $i$ such that $\pi_i - \pi_{i+1} > 2$.
For the number of descents, see St000021The number of descents of a permutation. and for the number of big descents, see St000647The number of big descents of a permutation.. General $r$-descents were for example be studied in [1, Section 2].
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
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
first row removal
Description
Removes the first entry of an integer partition
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