- FindStat

Identifier

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001087: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 138 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 14$

$F_{5} = 41 + q$

Description

The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern

$123$ , where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.

Map

inverse zeta map

Description

The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.

Map

Clarke-Steingrimsson-Zeng inverse

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map

$\Phi$ in [1, sec.3].

Map

Ringel

Description

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