Identifier
Values
[1,0] => [1] => 0
[1,0,1,0] => [1,2] => 0
[1,1,0,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => 1
[1,1,0,0,1,0] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => 2
[1,1,1,0,0,0] => [3,1,2] => 3
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0] => [1,4,2,3] => 3
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0] => [2,4,1,3] => 4
[1,1,1,0,0,0,1,0] => [3,1,2,4] => 3
[1,1,1,0,0,1,0,0] => [3,1,4,2] => 4
[1,1,1,0,1,0,0,0] => [3,4,1,2] => 4
[1,1,1,1,0,0,0,0] => [4,1,2,3] => 6
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 6
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 4
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => 5
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => 4
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => 5
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => 5
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => 7
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 4
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => 4
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 5
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => 6
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => 4
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => 5
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => 7
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => 7
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 6
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 7
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => 6
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => 8
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 10
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => 7
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => 7
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => 7
>>> Load all 196 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,2,3,4,6] => 6
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,6,4] => 7
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,2,6,3,4] => 6
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,2,3,4] => 8
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => 10
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => 3
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,4,5] => 4
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 3
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 3
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => 4
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,3,5] => 5
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,3,4,6] => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,3,6,4] => 5
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,3,4] => 5
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,3,4,5] => 7
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => 3
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => 3
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 4
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,4,5] => 5
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => 3
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => 4
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 4
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 5
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,1,5] => 6
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,1,4,6] => 5
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,1,6,4] => 6
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,1,4] => 6
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,1,4,5] => 8
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,1,3,5,6] => 4
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,1,3,6,5] => 5
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,1,5,3,6] => 5
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,1,5,6,3] => 6
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,1,6,3,5] => 7
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,1,3,6] => 5
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,1,6,3] => 6
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,1,3] => 8
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,1,3,5] => 8
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,1,3,4,6] => 7
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,1,3,6,4] => 8
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,1,6,3,4] => 7
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,1,3,4] => 9
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,1,3,4,5] => 11
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,2,4,5,6] => 3
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2,4,6,5] => 4
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,1,2,5,4,6] => 4
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,1,2,5,6,4] => 5
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,1,2,6,4,5] => 6
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => 4
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,1,4,2,6,5] => 5
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,1,4,5,2,6] => 5
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => 6
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,1,4,6,2,5] => 7
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,1,5,2,4,6] => 6
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,1,5,2,6,4] => 7
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,1,5,6,2,4] => 7
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,1,6,2,4,5] => 9
[1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,1,2,5,6] => 4
[1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,1,2,6,5] => 5
[1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,1,5,2,6] => 5
[1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => 6
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,1,6,2,5] => 7
[1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,1,2,6] => 7
[1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => 8
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => 8
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => 10
[1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,1,2,4,6] => 7
[1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => 8
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,1,6,2,4] => 7
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => 9
[1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,1,2,4,5] => 11
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,2,3,5,6] => 6
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,1,2,3,6,5] => 7
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,1,2,5,3,6] => 7
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => 8
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,1,2,6,3,5] => 9
[1,1,1,1,0,0,1,0,0,0,1,0] => [4,1,5,2,3,6] => 6
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,1,5,2,6,3] => 7
[1,1,1,1,0,0,1,0,1,0,0,0] => [4,1,5,6,2,3] => 10
[1,1,1,1,0,0,1,1,0,0,0,0] => [4,1,6,2,3,5] => 9
[1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,1,2,3,6] => 8
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => 9
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,1,6,2,3] => 10
[1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,1,2,3] => 9
[1,1,1,1,0,1,1,0,0,0,0,0] => [4,6,1,2,3,5] => 12
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,6] => 10
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => 11
[1,1,1,1,1,0,0,0,1,0,0,0] => [5,1,2,6,3,4] => 9
[1,1,1,1,1,0,0,1,0,0,0,0] => [5,1,6,2,3,4] => 11
[1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,1,2,3,4] => 12
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => 15
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
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Description
The sorting index of a permutation.
The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols.
Compare this to St000018The number of inversions of a permutation., the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.