- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000374: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 181 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 7\ q + 5\ q^{2} + q^{3}$

$F_{5} = 1 + 16\ q + 17\ q^{2} + 7\ q^{3} + q^{4}$

Description

The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ .
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let

$\pi$ be a permutation. Following Viennot [1], we associate to

$\pi$ a heap of pieces, by considering each decreasing run

$(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of

$\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.