Processing math: 100%

Identifier
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] => 1
[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] => 1
[1,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[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] => 1
[2,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[2,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[2,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[1,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[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] => 1
[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] => 2
[3,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[3,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[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] => 2
[2,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[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] => 1
[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] => 2
[4,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[4,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[3,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[3,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[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] => 2
[3,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[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] => 1
[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] => 2
[5,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[5,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[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] => 2
[4,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[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] => 2
[4,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[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] => 1
[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] => 2
[6,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[6,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[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] => 2
[5,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[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] => 2
[5,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[9,2] => [2] => [[1,2]] => [1,2] => 0
[9,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[8,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[8,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[8,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[7,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[7,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[7,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[7,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[7,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[6,5] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[6,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[6,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[6,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[6,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[6,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 2
[6,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[10,2] => [2] => [[1,2]] => [1,2] => 0
[10,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[9,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[9,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[9,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[8,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
>>> Load all 197 entries. <<<
[8,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[8,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[8,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[8,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[7,5] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[7,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[7,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[7,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[7,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[7,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 2
[7,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[11,2] => [2] => [[1,2]] => [1,2] => 0
[11,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[10,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[10,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[10,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[9,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[9,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[9,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[9,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[9,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[8,5] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[8,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[8,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[8,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[8,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[8,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 2
[8,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[12,2] => [2] => [[1,2]] => [1,2] => 0
[12,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[11,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[11,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[11,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[10,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[10,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[10,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[10,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[10,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[9,5] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[9,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[9,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[9,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[9,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[9,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 2
[9,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[13,2] => [2] => [[1,2]] => [1,2] => 0
[13,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[12,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[12,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[12,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[11,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[11,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[11,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[11,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[11,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[10,5] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[10,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[10,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[10,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[10,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[10,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 2
[10,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[14,2] => [2] => [[1,2]] => [1,2] => 0
[14,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[13,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[13,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[13,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[12,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[12,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[12,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[12,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[12,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[11,5] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[11,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[11,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[11,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[11,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[11,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 2
[11,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[15,2] => [2] => [[1,2]] => [1,2] => 0
[15,1,1] => [1,1] => [[1],[2]] => [2,1] => 0
[14,3] => [3] => [[1,2,3]] => [1,2,3] => 0
[14,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[14,1,1,1] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[13,4] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[13,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[13,2,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[13,2,1,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 1
[13,1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[12,5] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[12,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[12,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[12,3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1
[12,2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[12,2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 2
[12,1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
search for individual values
searching the database for the individual values of this statistic
Description
The number of inversions of the second entry of a permutation.
This is, for a permutation π of length n,
#{2<k≤n∣π(2)>π(k)}.
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Map
first row removal
Description
Removes the first entry of an integer partition
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.