- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 247 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[] => [] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{3} = q + q^{2} + q^{4}$

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

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

$F_{6} = 4\ q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5} + q^{7}$

Description

The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

Lehmer code rotation

Description

Sends a permutation

$\pi$ to the unique permutation

$\tau$ (of the same length) such that every entry in the Lehmer code of

$\tau$ is cyclically one larger than the Lehmer code of

$\pi$ .