- FindStat

Identifier

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001264: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 10 + 3\ q + q^{2}$

$F_{5} = 28 + 9\ q + 4\ q^{2} + q^{3}$

$F_{6} = 84 + 28\ q + 14\ q^{2} + 5\ q^{3} + q^{4}$

Description

The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

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

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 (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength

$n$ in an

$n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.