- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000374: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 131 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = q + 15\ q^{2} + 18\ q^{3} + 7\ q^{4} + q^{5}$

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

inverse

Description

Sends a permutation to its inverse.

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

Map

Ringel

Description

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