- FindStat

Identifier

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000672: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 2\ q + 8\ q^{2} + 4\ q^{3}$

$F_{5} = 3\ q + 14\ q^{2} + 19\ q^{3} + 6\ q^{4}$

Description

The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is

$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$
for some

$(r,a,b)$ .
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

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

Delest-Viennot-inverse

Description

Return the Dyck path obtained by applying the inverse of Delest-Viennot's bijection to the corresponding parallelogram polyomino.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
The Delest-Viennot bijection

$\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path

$(\beta^{(-1)}\circ\gamma)(D)$ .

Map

Ringel

Description

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