Identifier
Values
[1] => [[1]] => [1] => 0
[2] => [[1,2]] => [1,2] => 0
[1,1] => [[1],[2]] => [2,1] => 0
[3] => [[1,2,3]] => [1,2,3] => 0
[2,1] => [[1,3],[2]] => [2,1,3] => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => 0
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => 1
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 1
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 0
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => 1
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => 1
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 1
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 0
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 2
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 1
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 0
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 1
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 0
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => 0
[6,1] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => 1
[5,2] => [[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => 1
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => 1
[4,3] => [[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => 1
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [4,2,5,1,3,6,7] => 2
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => 1
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [5,2,6,7,1,3,4] => 1
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [5,6,3,4,1,2,7] => 1
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [5,3,2,6,1,4,7] => 2
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => 1
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3] => 1
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5] => 1
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => 1
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => 0
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => 0
[7,1] => [[1,3,4,5,6,7,8],[2]] => [2,1,3,4,5,6,7,8] => 1
[6,2] => [[1,2,5,6,7,8],[3,4]] => [3,4,1,2,5,6,7,8] => 1
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [3,2,1,4,5,6,7,8] => 1
[5,3] => [[1,2,3,7,8],[4,5,6]] => [4,5,6,1,2,3,7,8] => 1
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [4,2,5,1,3,6,7,8] => 2
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [4,3,2,1,5,6,7,8] => 1
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => 0
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [5,2,6,7,1,3,4,8] => 2
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [5,6,3,4,1,2,7,8] => 1
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [5,3,2,6,1,4,7,8] => 2
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => 1
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [6,7,3,4,8,1,2,5] => 1
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5] => 1
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3,8] => 2
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5,8] => 2
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => 1
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => 0
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4] => 1
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6] => 1
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => 1
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => 0
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => 0
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [2,1,3,4,5,6,7,8,9] => 1
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [3,4,1,2,5,6,7,8,9] => 1
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [3,2,1,4,5,6,7,8,9] => 1
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [4,5,6,1,2,3,7,8,9] => 1
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [4,2,5,1,3,6,7,8,9] => 2
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9] => 1
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [5,6,7,8,1,2,3,4,9] => 1
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [5,2,6,7,1,3,4,8,9] => 2
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [5,6,3,4,1,2,7,8,9] => 1
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [5,3,2,6,1,4,7,8,9] => 2
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9] => 1
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [6,2,7,8,9,1,3,4,5] => 1
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [6,7,3,4,8,1,2,5,9] => 2
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5,9] => 2
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3,8,9] => 2
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5,8,9] => 2
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9] => 1
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => 0
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [7,4,8,2,5,9,1,3,6] => 2
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [7,4,3,2,8,9,1,5,6] => 1
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2,9] => 1
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4,9] => 2
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6,9] => 2
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9] => 1
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [8,6,9,4,7,2,5,1,3] => 1
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [8,6,4,3,9,2,7,1,5] => 1
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [8,6,5,4,3,2,9,1,7] => 1
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9] => 1
[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] => 0
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => 0
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [2,1,3,4,5,6,7,8,9,10] => 1
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [3,4,1,2,5,6,7,8,9,10] => 1
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [3,2,1,4,5,6,7,8,9,10] => 1
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [4,5,6,1,2,3,7,8,9,10] => 1
>>> Load all 141 entries. <<<
[7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => [4,2,5,1,3,6,7,8,9,10] => 2
[7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9,10] => 1
[6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => [5,6,7,8,1,2,3,4,9,10] => 1
[6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => [5,2,6,7,1,3,4,8,9,10] => 2
[6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => [5,6,3,4,1,2,7,8,9,10] => 1
[6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => [5,3,2,6,1,4,7,8,9,10] => 2
[6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9,10] => 1
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [6,7,8,9,10,1,2,3,4,5] => 0
[5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => [6,2,7,8,9,1,3,4,5,10] => 2
[5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => [6,7,3,4,8,1,2,5,9,10] => 2
[5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5,9,10] => 2
[5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3,8,9,10] => 2
[5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5,8,9,10] => 2
[5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9,10] => 1
[4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => [7,8,3,4,9,10,1,2,5,6] => 1
[4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => [7,3,2,8,9,10,1,4,5,6] => 1
[4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3,10] => 1
[4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => [7,4,8,2,5,9,1,3,6,10] => 3
[4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => [7,4,3,2,8,9,1,5,6,10] => 2
[4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2,9,10] => 1
[4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4,9,10] => 2
[4,2,1,1,1,1] => [[1,6,9,10],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6,9,10] => 2
[4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9,10] => 1
[3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => [8,5,9,10,2,6,7,1,3,4] => 1
[3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => [8,9,5,6,3,4,10,1,2,7] => 1
[3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => [8,5,3,9,2,6,10,1,4,7] => 2
[3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => [8,5,4,3,2,9,10,1,6,7] => 1
[3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => [8,6,9,4,7,2,5,1,3,10] => 2
[3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => [8,6,4,3,9,2,7,1,5,10] => 2
[3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => [8,6,5,4,3,2,9,1,7,10] => 2
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9,10] => 1
[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] => 0
[2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => [9,7,5,10,3,8,2,6,1,4] => 1
[2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => [9,7,5,4,3,10,2,8,1,6] => 1
[2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => [9,7,6,5,4,3,2,10,1,8] => 1
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1,10] => 1
[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] => 0
[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] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [12,11,10,9,8,7,6,5,4,3,2,1] => 0
[] => [] => [] => 0
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Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the 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
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.