- FindStat

Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 14 + 14\ q + 9\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 42 + 42\ q + 28\ q^{2} + 14\ q^{3} + 5\ q^{4} + q^{5}$

Description

The bounce statistic of a Dyck path.
The bounce path

$D'$ of a Dyck path

$D$ is the Dyck path obtained from

$D$ by starting at the end point

$(2n,0)$ , traveling north-west until hitting

$D$ , then bouncing back south-west to the

$x$ -axis, and repeating this procedure until finally reaching the point

$(0,0)$ .
The points where

$D'$ touches the

$x$ -axis are called bounce points, and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all

$i$ for which the bounce path

$D'$ of

$D$ touches the

$x$ -axis at

$(2i,0)$ .
In particular, the bounce statistics of

$D$ and

$D'$ coincide.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.