- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤ

Values

[1] => [1,0] => [1] => [[1]] => 0

[1,1] => [1,0,1,0] => [2,1] => [[1],[2]] => 1

[2] => [1,1,0,0] => [1,2] => [[1,2]] => 0

[1,1,1] => [1,0,1,0,1,0] => [3,2,1] => [[1],[2],[3]] => 3

[1,2] => [1,0,1,1,0,0] => [2,3,1] => [[1,3],[2]] => 1

[2,1] => [1,1,0,0,1,0] => [3,1,2] => [[1,2],[3]] => 2

[3] => [1,1,1,0,0,0] => [1,2,3] => [[1,2,3]] => 0

[1,1,1,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => [[1],[2],[3],[4]] => 6

[1,1,2] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => [[1,4],[2],[3]] => 3

[1,2,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => [[1,3],[2],[4]] => 4

[1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [[1,3,4],[2]] => 1

[2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => [[1,2],[3],[4]] => 5

[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [[1,2],[3,4]] => 2

[3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [[1,2,3],[4]] => 3

[4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [[1,2,3,4]] => 0

[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]] => 10

[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [[1,5],[2],[3],[4]] => 6

[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [[1,4],[2],[3],[5]] => 7

[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [[1,4,5],[2],[3]] => 3

[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [[1,3],[2],[4],[5]] => 8

[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [[1,3],[2,5],[4]] => 4

[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [[1,3,4],[2],[5]] => 5

[1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [[1,3,4,5],[2]] => 1

[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [[1,2],[3],[4],[5]] => 9

[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [[1,2],[3,5],[4]] => 5

[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [[1,2],[3,4],[5]] => 6

[2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [[1,2,5],[3,4]] => 2

[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [[1,2,3],[4],[5]] => 7

[3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [[1,2,3],[4,5]] => 3

[4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [[1,2,3,4],[5]] => 4

[5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [[1,2,3,4,5]] => 0

[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]] => 15

[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5]] => 10

[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [[1,5],[2],[3],[4],[6]] => 11

[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [[1,5,6],[2],[3],[4]] => 6

[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]] => 12

[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [[1,4],[2,6],[3],[5]] => 7

[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [[1,4,5],[2],[3],[6]] => 8

[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [[1,4,5,6],[2],[3]] => 3

[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [[1,3],[2],[4],[5],[6]] => 13

[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [[1,3],[2,6],[4],[5]] => 8

[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]] => 9

[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [[1,3,6],[2,5],[4]] => 4

[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [[1,3,4],[2],[5],[6]] => 10

[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [[1,3,4],[2,6],[5]] => 5

[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [[1,3,4,5],[2],[6]] => 6

[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [[1,3,4,5,6],[2]] => 1

[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => [[1,2],[3],[4],[5],[6]] => 14

[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => [[1,2],[3,6],[4],[5]] => 9

[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => [[1,2],[3,5],[4],[6]] => 10

[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => [[1,2,6],[3,5],[4]] => 5

[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => [[1,2],[3,4],[5],[6]] => 11

[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [[1,2],[3,4],[5,6]] => 6

[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [[1,2,5],[3,4],[6]] => 7

[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [[1,2,5,6],[3,4]] => 2

[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => [[1,2,3],[4],[5],[6]] => 12

[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => [[1,2,3],[4,6],[5]] => 7

[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => [[1,2,3],[4,5],[6]] => 8

[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [[1,2,3],[4,5,6]] => 3

[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => [[1,2,3,4],[5],[6]] => 9

[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [[1,2,3,4],[5,6]] => 4

[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [[1,2,3,4,5],[6]] => 5

[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [[1,2,3,4,5,6]] => 0

[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7]] => 21

[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [6,7,5,4,3,2,1] => [[1,7],[2],[3],[4],[5],[6]] => 15

[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7]] => 16

[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [5,6,7,4,3,2,1] => [[1,6,7],[2],[3],[4],[5]] => 10

[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [7,6,4,5,3,2,1] => [[1,5],[2],[3],[4],[6],[7]] => 17

[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [6,7,4,5,3,2,1] => [[1,5],[2,7],[3],[4],[6]] => 11

[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [7,4,5,6,3,2,1] => [[1,5,6],[2],[3],[4],[7]] => 12

[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [4,5,6,7,3,2,1] => [[1,5,6,7],[2],[3],[4]] => 6

[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [7,6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6],[7]] => 18

[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [6,7,5,3,4,2,1] => [[1,4],[2,7],[3],[5],[6]] => 12

[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [7,5,6,3,4,2,1] => [[1,4],[2,6],[3],[5],[7]] => 13

[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [5,6,7,3,4,2,1] => [[1,4,7],[2,6],[3],[5]] => 7

[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [7,6,3,4,5,2,1] => [[1,4,5],[2],[3],[6],[7]] => 14

[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [6,7,3,4,5,2,1] => [[1,4,5],[2,7],[3],[6]] => 8

[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [7,3,4,5,6,2,1] => [[1,4,5,6],[2],[3],[7]] => 9

[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,2,1] => [[1,4,5,6,7],[2],[3]] => 3

[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [7,6,5,4,2,3,1] => [[1,3],[2],[4],[5],[6],[7]] => 19

[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [6,7,5,4,2,3,1] => [[1,3],[2,7],[4],[5],[6]] => 13

[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [7,5,6,4,2,3,1] => [[1,3],[2,6],[4],[5],[7]] => 14

[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [5,6,7,4,2,3,1] => [[1,3,7],[2,6],[4],[5]] => 8

[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,2,3,1] => [[1,3],[2,5],[4],[6],[7]] => 15

[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,2,3,1] => [[1,3],[2,5],[4,7],[6]] => 9

[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,2,3,1] => [[1,3,6],[2,5],[4],[7]] => 10

[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,2,3,1] => [[1,3,6,7],[2,5],[4]] => 4

[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [7,6,5,2,3,4,1] => [[1,3,4],[2],[5],[6],[7]] => 16

[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [6,7,5,2,3,4,1] => [[1,3,4],[2,7],[5],[6]] => 10

[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [7,5,6,2,3,4,1] => [[1,3,4],[2,6],[5],[7]] => 11

[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [5,6,7,2,3,4,1] => [[1,3,4],[2,6,7],[5]] => 5

[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [7,6,2,3,4,5,1] => [[1,3,4,5],[2],[6],[7]] => 12

[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [6,7,2,3,4,5,1] => [[1,3,4,5],[2,7],[6]] => 6

[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [7,2,3,4,5,6,1] => [[1,3,4,5,6],[2],[7]] => 7

[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [[1,3,4,5,6,7],[2]] => 1

[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,1,2] => [[1,2],[3],[4],[5],[6],[7]] => 20

[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [6,7,5,4,3,1,2] => [[1,2],[3,7],[4],[5],[6]] => 14

[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [7,5,6,4,3,1,2] => [[1,2],[3,6],[4],[5],[7]] => 15

[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [5,6,7,4,3,1,2] => [[1,2,7],[3,6],[4],[5]] => 9

[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [7,6,4,5,3,1,2] => [[1,2],[3,5],[4],[6],[7]] => 16

[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [6,7,4,5,3,1,2] => [[1,2],[3,5],[4,7],[6]] => 10

>>> Load all 286 entries. <<<

[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [7,4,5,6,3,1,2] => [[1,2,6],[3,5],[4],[7]] => 11

[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [4,5,6,7,3,1,2] => [[1,2,6,7],[3,5],[4]] => 5

[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [7,6,5,3,4,1,2] => [[1,2],[3,4],[5],[6],[7]] => 17

[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [6,7,5,3,4,1,2] => [[1,2],[3,4],[5,7],[6]] => 11

[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [7,5,6,3,4,1,2] => [[1,2],[3,4],[5,6],[7]] => 12

[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [5,6,7,3,4,1,2] => [[1,2,7],[3,4],[5,6]] => 6

[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [7,6,3,4,5,1,2] => [[1,2,5],[3,4],[6],[7]] => 13

[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [6,7,3,4,5,1,2] => [[1,2,5],[3,4],[6,7]] => 7

[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [7,3,4,5,6,1,2] => [[1,2,5,6],[3,4],[7]] => 8

[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,1,2] => [[1,2,5,6,7],[3,4]] => 2

[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [7,6,5,4,1,2,3] => [[1,2,3],[4],[5],[6],[7]] => 18

[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [6,7,5,4,1,2,3] => [[1,2,3],[4,7],[5],[6]] => 12

[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [7,5,6,4,1,2,3] => [[1,2,3],[4,6],[5],[7]] => 13

[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [5,6,7,4,1,2,3] => [[1,2,3],[4,6,7],[5]] => 7

[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => [[1,2,3],[4,5],[6],[7]] => 14

[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [[1,2,3],[4,5],[6,7]] => 8

[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [[1,2,3],[4,5,6],[7]] => 9

[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [[1,2,3,7],[4,5,6]] => 3

[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [7,6,5,1,2,3,4] => [[1,2,3,4],[5],[6],[7]] => 15

[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [6,7,5,1,2,3,4] => [[1,2,3,4],[5,7],[6]] => 9

[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [7,5,6,1,2,3,4] => [[1,2,3,4],[5,6],[7]] => 10

[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [5,6,7,1,2,3,4] => [[1,2,3,4],[5,6,7]] => 4

[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [7,6,1,2,3,4,5] => [[1,2,3,4,5],[6],[7]] => 11

[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [6,7,1,2,3,4,5] => [[1,2,3,4,5],[6,7]] => 5

[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7]] => 6

[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7]] => 0

[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 28

[1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,3,2,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => 21

[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [8,6,7,5,4,3,2,1] => [[1,7],[2],[3],[4],[5],[6],[8]] => 22

[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [6,7,8,5,4,3,2,1] => [[1,7,8],[2],[3],[4],[5],[6]] => 15

[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8]] => 23

[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [7,8,5,6,4,3,2,1] => [[1,6],[2,8],[3],[4],[5],[7]] => 16

[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [8,5,6,7,4,3,2,1] => [[1,6,7],[2],[3],[4],[5],[8]] => 17

[1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [5,6,7,8,4,3,2,1] => [[1,6,7,8],[2],[3],[4],[5]] => 10

[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [8,7,6,4,5,3,2,1] => [[1,5],[2],[3],[4],[6],[7],[8]] => 24

[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [7,8,6,4,5,3,2,1] => [[1,5],[2,8],[3],[4],[6],[7]] => 17

[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [8,6,7,4,5,3,2,1] => [[1,5],[2,7],[3],[4],[6],[8]] => 18

[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [8,7,4,5,6,3,2,1] => [[1,5,6],[2],[3],[4],[7],[8]] => 19

[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [8,4,5,6,7,3,2,1] => [[1,5,6,7],[2],[3],[4],[8]] => 13

[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [4,5,6,7,8,3,2,1] => [[1,5,6,7,8],[2],[3],[4]] => 6

[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6],[7],[8]] => 25

[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,3,4,2,1] => [[1,4],[2,8],[3],[5],[6],[7]] => 18

[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,3,4,2,1] => [[1,4],[2,6],[3],[5],[7],[8]] => 20

[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,3,4,2,1] => [[1,4],[2,6],[3,8],[5],[7]] => 13

[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,3,4,2,1] => [[1,4,7],[2,6],[3],[5],[8]] => 14

[1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,3,4,2,1] => [[1,4,7,8],[2,6],[3],[5]] => 7

[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [8,7,6,3,4,5,2,1] => [[1,4,5],[2],[3],[6],[7],[8]] => 21

[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [7,8,6,3,4,5,2,1] => [[1,4,5],[2,8],[3],[6],[7]] => 14

[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [8,6,7,3,4,5,2,1] => [[1,4,5],[2,7],[3],[6],[8]] => 15

[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [6,7,8,3,4,5,2,1] => [[1,4,5],[2,7,8],[3],[6]] => 8

[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [8,7,3,4,5,6,2,1] => [[1,4,5,6],[2],[3],[7],[8]] => 16

[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [7,8,3,4,5,6,2,1] => [[1,4,5,6],[2,8],[3],[7]] => 9

[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [8,3,4,5,6,7,2,1] => [[1,4,5,6,7],[2],[3],[8]] => 10

[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,4,5,6,7,8,2,1] => [[1,4,5,6,7,8],[2],[3]] => 3

[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,2,3,1] => [[1,3],[2],[4],[5],[6],[7],[8]] => 26

[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,2,3,1] => [[1,3],[2,8],[4],[5],[6],[7]] => 19

[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [8,6,7,5,4,2,3,1] => [[1,3],[2,7],[4],[5],[6],[8]] => 20

[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [8,7,5,6,4,2,3,1] => [[1,3],[2,6],[4],[5],[7],[8]] => 21

[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [8,5,6,7,4,2,3,1] => [[1,3,7],[2,6],[4],[5],[8]] => 15

[1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [5,6,7,8,4,2,3,1] => [[1,3,7,8],[2,6],[4],[5]] => 8

[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [8,7,6,4,5,2,3,1] => [[1,3],[2,5],[4],[6],[7],[8]] => 22

[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [8,6,7,4,5,2,3,1] => [[1,3],[2,5],[4,7],[6],[8]] => 16

[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [8,7,4,5,6,2,3,1] => [[1,3,6],[2,5],[4],[7],[8]] => 17

[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [7,8,4,5,6,2,3,1] => [[1,3,6],[2,5],[4,8],[7]] => 10

[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [8,4,5,6,7,2,3,1] => [[1,3,6,7],[2,5],[4],[8]] => 11

[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [4,5,6,7,8,2,3,1] => [[1,3,6,7,8],[2,5],[4]] => 4

[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,2,3,4,1] => [[1,3,4],[2],[5],[6],[7],[8]] => 23

[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [8,6,7,5,2,3,4,1] => [[1,3,4],[2,7],[5],[6],[8]] => 17

[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,2,3,4,1] => [[1,3,4],[2,7,8],[5],[6]] => 10

[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,2,3,4,1] => [[1,3,4],[2,6],[5],[7],[8]] => 18

[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,2,3,4,1] => [[1,3,4],[2,6],[5,8],[7]] => 11

[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,2,3,4,1] => [[1,3,4],[2,6,7],[5],[8]] => 12

[1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,2,3,4,1] => [[1,3,4,8],[2,6,7],[5]] => 5

[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [8,7,6,2,3,4,5,1] => [[1,3,4,5],[2],[6],[7],[8]] => 19

[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [7,8,6,2,3,4,5,1] => [[1,3,4,5],[2,8],[6],[7]] => 12

[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [8,6,7,2,3,4,5,1] => [[1,3,4,5],[2,7],[6],[8]] => 13

[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [6,7,8,2,3,4,5,1] => [[1,3,4,5],[2,7,8],[6]] => 6

[1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [8,7,2,3,4,5,6,1] => [[1,3,4,5,6],[2],[7],[8]] => 14

[1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [7,8,2,3,4,5,6,1] => [[1,3,4,5,6],[2,8],[7]] => 7

[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [8,2,3,4,5,6,7,1] => [[1,3,4,5,6,7],[2],[8]] => 8

[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [[1,3,4,5,6,7,8],[2]] => 1

[2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,3,1,2] => [[1,2],[3],[4],[5],[6],[7],[8]] => 27

[2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,3,1,2] => [[1,2],[3,8],[4],[5],[6],[7]] => 20

[2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [8,6,7,5,4,3,1,2] => [[1,2],[3,7],[4],[5],[6],[8]] => 21

[2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [6,7,8,5,4,3,1,2] => [[1,2,8],[3,7],[4],[5],[6]] => 14

[2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [8,7,5,6,4,3,1,2] => [[1,2],[3,6],[4],[5],[7],[8]] => 22

[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0] => [7,8,5,6,4,3,1,2] => [[1,2],[3,6],[4,8],[5],[7]] => 15

[2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0] => [5,6,7,8,4,3,1,2] => [[1,2,7,8],[3,6],[4],[5]] => 9

[2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0] => [8,7,6,4,5,3,1,2] => [[1,2],[3,5],[4],[6],[7],[8]] => 23

[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => [7,8,6,4,5,3,1,2] => [[1,2],[3,5],[4,8],[6],[7]] => 16

[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0] => [8,6,7,4,5,3,1,2] => [[1,2],[3,5],[4,7],[6],[8]] => 17

[2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0] => [6,7,8,4,5,3,1,2] => [[1,2,8],[3,5],[4,7],[6]] => 10

[2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [4,5,6,7,8,3,1,2] => [[1,2,6,7,8],[3,5],[4]] => 5

[2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,3,4,1,2] => [[1,2],[3,4],[5],[6],[7],[8]] => 24

[2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,3,4,1,2] => [[1,2],[3,4],[5,8],[6],[7]] => 17

[2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,3,4,1,2] => [[1,2,8],[3,4],[5,7],[6]] => 11

[2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,3,4,1,2] => [[1,2],[3,4],[5,6],[7],[8]] => 19

[2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,3,4,1,2] => [[1,2],[3,4],[5,6],[7,8]] => 12

[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,3,4,1,2] => [[1,2,7],[3,4],[5,6],[8]] => 13

[2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,3,4,1,2] => [[1,2,7,8],[3,4],[5,6]] => 6

[2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [8,7,6,3,4,5,1,2] => [[1,2,5],[3,4],[6],[7],[8]] => 20

[2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0] => [8,6,7,3,4,5,1,2] => [[1,2,5],[3,4],[6,7],[8]] => 14

[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0] => [8,7,3,4,5,6,1,2] => [[1,2,5,6],[3,4],[7],[8]] => 15

[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [7,8,3,4,5,6,1,2] => [[1,2,5,6],[3,4],[7,8]] => 8

[2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [8,3,4,5,6,7,1,2] => [[1,2,5,6,7],[3,4],[8]] => 9

[2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,4,5,6,7,8,1,2] => [[1,2,5,6,7,8],[3,4]] => 2

[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,1,2,3] => [[1,2,3],[4],[5],[6],[7],[8]] => 25

[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,1,2,3] => [[1,2,3],[4,8],[5],[6],[7]] => 18

[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0] => [6,7,8,5,4,1,2,3] => [[1,2,3],[4,7,8],[5],[6]] => 12

[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0] => [7,8,5,6,4,1,2,3] => [[1,2,3],[4,6],[5,8],[7]] => 13

[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0] => [8,5,6,7,4,1,2,3] => [[1,2,3],[4,6,7],[5],[8]] => 14

[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0] => [5,6,7,8,4,1,2,3] => [[1,2,3,8],[4,6,7],[5]] => 7

[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0] => [8,6,7,4,5,1,2,3] => [[1,2,3],[4,5],[6,7],[8]] => 15

[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0] => [6,7,8,4,5,1,2,3] => [[1,2,3],[4,5,8],[6,7]] => 8

[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0] => [8,7,4,5,6,1,2,3] => [[1,2,3],[4,5,6],[7],[8]] => 16

[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => [8,4,5,6,7,1,2,3] => [[1,2,3,7],[4,5,6],[8]] => 10

[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [4,5,6,7,8,1,2,3] => [[1,2,3,7,8],[4,5,6]] => 3

[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,1,2,3,4] => [[1,2,3,4],[5],[6],[7],[8]] => 22

[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,1,2,3,4] => [[1,2,3,4],[5,8],[6],[7]] => 15

[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0] => [8,6,7,5,1,2,3,4] => [[1,2,3,4],[5,7],[6],[8]] => 16

[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,1,2,3,4] => [[1,2,3,4],[5,7,8],[6]] => 9

[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,1,2,3,4] => [[1,2,3,4],[5,6],[7],[8]] => 17

[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,1,2,3,4] => [[1,2,3,4],[5,6],[7,8]] => 10

[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,1,2,3,4] => [[1,2,3,4],[5,6,7],[8]] => 11

[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,1,2,3,4] => [[1,2,3,4],[5,6,7,8]] => 4

[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => [8,7,6,1,2,3,4,5] => [[1,2,3,4,5],[6],[7],[8]] => 18

[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0] => [7,8,6,1,2,3,4,5] => [[1,2,3,4,5],[6,8],[7]] => 11

[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0] => [8,6,7,1,2,3,4,5] => [[1,2,3,4,5],[6,7],[8]] => 12

[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [6,7,8,1,2,3,4,5] => [[1,2,3,4,5],[6,7,8]] => 5

[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [8,7,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7],[8]] => 13

[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [7,8,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7,8]] => 6

[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7],[8]] => 7

[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8] => [[1,2,3,4,5,6,7,8]] => 0

[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 36

[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [8,9,7,6,5,4,3,2,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => 28

[1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [7,8,9,6,5,4,3,2,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => 21

[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [8,9,6,7,5,4,3,2,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => 22

[1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [4,5,6,7,8,9,3,2,1] => [[1,5,6,7,8,9],[2],[3],[4]] => 6

[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [3,4,5,6,7,8,9,2,1] => [[1,4,5,6,7,8,9],[2],[3]] => 3

[1,2,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [8,9,6,7,4,5,2,3,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => 16

[1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [4,5,6,7,8,9,2,3,1] => [[1,3,6,7,8,9],[2,5],[4]] => 4

[1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [6,7,8,9,2,3,4,5,1] => [[1,3,4,5],[2,7,8,9],[6]] => 6

[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [[1,3,4,5,6,7,8,9],[2]] => 1

[2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,4,3,1,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => 35

[2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [9,7,8,5,6,3,4,1,2] => [[1,2],[3,4],[5,6],[7,8],[9]] => 20

[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [3,4,5,6,7,8,9,1,2] => [[1,2,5,6,7,8,9],[3,4]] => 2

[3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,4,1,2,3] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => 33

[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [4,5,6,7,8,9,1,2,3] => [[1,2,3,7,8,9],[4,5,6]] => 3

[4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,1,2,3,4] => [[1,2,3,4],[5],[6],[7],[8],[9]] => 30

[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [5,6,7,8,9,1,2,3,4] => [[1,2,3,4,9],[5,6,7,8]] => 4

[5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0] => [9,8,7,6,1,2,3,4,5] => [[1,2,3,4,5],[6],[7],[8],[9]] => 26

[5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => [6,7,8,9,1,2,3,4,5] => [[1,2,3,4,5],[6,7,8,9]] => 5

[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0] => [9,8,7,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7],[8],[9]] => 21

[6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0] => [9,7,8,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7,8],[9]] => 14

[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0] => [7,8,9,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7,8,9]] => 6

[7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => [9,8,1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7],[8],[9]] => 15

[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [8,9,1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7],[8,9]] => 7

[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [9,1,2,3,4,5,6,7,8] => [[1,2,3,4,5,6,7,8],[9]] => 8

[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9]] => 0

[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 45

[1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,10,8,7,6,5,4,3,2,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => 36

[1,1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [6,7,8,9,10,5,4,3,2,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => 15

[1,1,1,7] => [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [4,5,6,7,8,9,10,3,2,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => 6

[1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [3,4,5,6,7,8,9,10,2,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => 3

[1,2,7] => [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [4,5,6,7,8,9,10,2,3,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => 4

[1,3,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [8,9,10,5,6,7,2,3,4,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => 12

[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [[1,3,4,5,6,7,8,9,10],[2]] => 1

[2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,5,4,3,1,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => 44

[2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [9,10,7,8,5,6,3,4,1,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => 20

[2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [3,4,5,6,7,8,9,10,1,2] => [[1,2,5,6,7,8,9,10],[3,4]] => 2

[3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,5,4,1,2,3] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => 42

[3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [4,5,6,7,8,9,10,1,2,3] => [[1,2,3,7,8,9,10],[4,5,6]] => 3

[4,1,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,5,1,2,3,4] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => 39

[4,6] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [5,6,7,8,9,10,1,2,3,4] => [[1,2,3,4,9,10],[5,6,7,8]] => 4

[5,1,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,1,2,3,4,5] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => 35

[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [6,7,8,9,10,1,2,3,4,5] => [[1,2,3,4,5],[6,7,8,9,10]] => 5

[6,1,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0] => [10,9,8,7,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => 30

[6,4] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7,8,9,10]] => 6

[7,1,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0] => [10,9,8,1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7],[8],[9],[10]] => 24

[7,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0] => [8,9,10,1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7],[8,9,10]] => 7

[8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0] => [10,9,1,2,3,4,5,6,7,8] => [[1,2,3,4,5,6,7,8],[9],[10]] => 17

[8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => [9,10,1,2,3,4,5,6,7,8] => [[1,2,3,4,5,6,7,8],[9,10]] => 8

[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9],[10]] => 9

[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10]] => 0

[2,2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [11,12,9,10,7,8,5,6,3,4,1,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => 30

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

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{2} + q^{3}$

$F_{4} = 1 + q + q^{2} + 2\ q^{3} + q^{4} + q^{5} + q^{6}$

$F_{5} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + 2\ q^{6} + 2\ q^{7} + q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 3\ q^{7} + 3\ q^{8} + 3\ q^{9} + 3\ q^{10} + 2\ q^{11} + 2\ q^{12} + q^{13} + q^{14} + q^{15}$

$F_{7} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 4\ q^{6} + 4\ q^{7} + 4\ q^{8} + 5\ q^{9} + 5\ q^{10} + 5\ q^{11} + 5\ q^{12} + 4\ q^{13} + 4\ q^{14} + 4\ q^{15} + 3\ q^{16} + 2\ q^{17} + 2\ q^{18} + q^{19} + q^{20} + q^{21}$

Description

The (standard) major index of a standard tableau.
A descent of a standard tableau

$T$ is an index

$i$ such that

$i+1$ appears in a row strictly below the row of

$i$ . The (standard) major index is the the sum of the descents.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

Robinson-Schensted insertion tableau

Description

Sends a permutation to its Robinson-Schensted insertion tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length

$n$ and pairs of standard Young tableaux of the same shape and of size

$n$ , see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding insertion tableau.