- FindStat

Your data matches 213 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000308
(load all 7 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.

Matching statistic: St000778

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000778: Graphs ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,1,0,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [5,2,1,3,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,1,6,5,2,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,4,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [5,3,1,2,7,6,4] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [5,4,1,2,7,6,3] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => [7,6,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [4,2,1,7,6,3,5] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2,1,7,6,3,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,5,4,1,6,7,3] => [4,2,1,7,6,5,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,0,1,0,0,0,1,1,0,0]
 => [6,3,4,1,2,7,5] => [4,3,1,7,6,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [5,3,4,1,6,7,2] => [4,3,1,7,6,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [6,3,2,1,7,4,5] => ([(0,1),(0,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [2,3,6,5,1,7,4] => [5,3,2,1,7,6,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,6,4,5,1,7,3] => [5,4,2,1,7,6,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,4,5,1,7,2] => [5,4,3,1,7,6,2] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,4,5}

Description

The metric dimension of a graph. This is the length of the shortest vector of vertices, such that every vertex is uniquely determined by the vector of distances from these vertices.

Matching statistic: St000996
(load all 12 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,6,4,5,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,6,4,3,5,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,6,3,5,4,2] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,6,3,4,5,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,4,3,6,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5,3,4,6,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,3,6,5,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,5,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,6,4,5,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,5,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,5,3,4,2,6] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,3,5,2,6] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,2,6] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,7,4,3,5,6,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,7,3,5,4,6,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,7,3,4,6,5,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,7,3,4,5,6,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,6,4,5,3,7,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,6,4,3,5,7,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,6,3,5,4,7,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,6,3,4,5,7,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,5,4,3,7,6,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,5,3,4,7,6,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,5,4,3,6,7,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,5,3,4,6,7,2] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,5,4,3,2,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,6,4,5,3,2,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,6,4,3,5,2,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,6,3,5,4,2,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,6,3,4,5,2,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,5,4,3,6,2,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,5,3,4,6,2,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,4,3,2,7,6] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,5,3,4,2,7,6] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,4,3,2,6,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,5,3,4,2,6,7] => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} - 1

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

Matching statistic: St001644
(load all 2 compositions to match this statistic)

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[1,1,0,0]
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,4,4} - 1
[1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,4,4} - 1
[1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,4,4} - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,5),(1,2),(1,3),(1,4),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5} - 1

Description

The dimension of a graph. The dimension of a graph is the least integer

$n$ such that there exists a representation of the graph in the Euclidean space of dimension

$n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.

Matching statistic: St000374
(load all 8 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [1,5,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [1,5,4,2,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [1,4,2,5,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [1,4,5,3,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [1,3,5,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,4,3,2,5] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,3,2,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,5,3,2,6,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,3,2,4,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,5,4,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,4,2,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,4,2,6,5,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,3,5,6,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,4,2,6,3,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,5,2,3,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,5,2,3,6,4] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,5,2,4,6,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,4,3,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,5,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,5,3,6,4] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,3,6,5,4,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,3,6,4,2,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,4,6,5,3,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,6,4,5,3,2] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,4,6,3,2,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,3,6,5,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,3,5,2,6,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,5,6,3,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,3,6,2,4,5] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => [1,6,7,5,4,3,2] => [1,7,5,4,3,2,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [1,5,7,6,4,3,2] => [1,7,6,4,3,2,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => [1,7,5,6,4,3,2] => [1,6,4,3,2,7,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => [1,5,6,7,4,3,2] => [1,7,4,3,2,5,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => [1,7,6,5,3,2,4] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => [1,4,6,7,5,3,2] => [1,7,5,3,2,4,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => [1,7,6,4,5,3,2] => [1,5,3,2,7,6,4] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,1,2,5,3,7,4] => [1,6,7,4,5,3,2] => [1,5,3,2,7,4,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [1,4,5,7,6,3,2] => [1,7,6,3,2,4,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => [1,4,7,5,6,3,2] => [1,6,3,2,4,7,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => [1,7,4,5,6,3,2] => [1,6,3,2,5,7,4] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [1,4,5,6,7,3,2] => [1,7,3,2,4,5,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => [1,7,6,5,4,2,3] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => [1,3,6,7,5,4,2] => [1,7,5,4,2,3,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [1,3,5,7,6,4,2] => [1,7,6,4,2,3,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => [1,3,7,5,6,4,2] => [1,6,4,2,3,7,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => [1,7,4,2,3,5,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [1,3,4,7,6,5,2] => [1,7,6,5,2,3,4] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [1,3,4,6,7,5,2] => [1,7,5,2,3,4,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [1,3,7,6,4,5,2] => [1,5,2,3,7,6,4] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [1,3,6,7,4,5,2] => [1,5,2,3,7,4,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [1,7,6,3,4,5,2] => [1,5,2,4,7,6,3] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [1,6,7,3,4,5,2] => [1,5,2,4,7,3,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [1,3,4,5,7,6,2] => [1,7,6,2,3,4,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [1,3,7,4,5,6,2] => [1,6,2,3,5,7,4] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [1,7,3,4,5,6,2] => [1,6,2,4,5,7,3] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [1,6,7,5,3,2,4] => [1,4,7,5,3,2,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [1,7,5,3,2,6,4] => [1,6,4,7,5,3,2] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [1,5,3,2,6,7,4] => [1,7,4,5,3,2,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [1,5,7,6,3,2,4] => [1,4,7,6,3,2,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [1,5,6,3,2,7,4] => [1,7,4,6,3,2,5] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => [1,4,7,3,2,5,6] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [7,4,1,6,2,3,5] => [1,7,5,2,4,6,3] => [1,6,3,4,7,5,2] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [1,6,3,2,7,4,5] => [1,5,7,4,6,3,2] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,7,5,1,2,3,4] => [1,6,3,5,2,7,4] => [1,7,4,5,2,6,3] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [1,7,3,4,5,2,6] => [1,5,2,4,6,7,3] => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1

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 [[St000213]] and [[St000119]].

Matching statistic: St000982
(load all 8 compositions to match this statistic)

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 10 => 01 => 01 => 1
[1,0,1,0]
 => 1010 => 0101 => 1001 => 2
[1,1,0,0]
 => 1100 => 1001 => 0101 => 1
[1,0,1,0,1,0]
 => 101010 => 010101 => 110001 => 3
[1,0,1,1,0,0]
 => 101100 => 011001 => 010101 => 1
[1,1,0,0,1,0]
 => 110010 => 100101 => 101001 => 2
[1,1,0,1,0,0]
 => 110100 => 101001 => 011001 => 2
[1,1,1,0,0,0]
 => 111000 => 110001 => 001101 => 2
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 11100001 => 4
[1,0,1,0,1,1,0,0]
 => 10101100 => 01011001 => 01100101 => 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 01100101 => 10101001 => 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 01101001 => 01101001 => 2
[1,0,1,1,1,0,0,0]
 => 10111000 => 01110001 => 00101101 => 2
[1,1,0,0,1,0,1,0]
 => 11001010 => 10010101 => 11010001 => 3
[1,1,0,0,1,1,0,0]
 => 11001100 => 10011001 => 01010101 => 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 10100101 => 10110001 => 3
[1,1,0,1,0,1,0,0]
 => 11010100 => 10101001 => 01110001 => 3
[1,1,0,1,1,0,0,0]
 => 11011000 => 10110001 => 00110101 => 2
[1,1,1,0,0,0,1,0]
 => 11100010 => 11000101 => 10011001 => 2
[1,1,1,0,0,1,0,0]
 => 11100100 => 11001001 => 01011001 => 2
[1,1,1,0,1,0,0,0]
 => 11101000 => 11010001 => 00111001 => 3
[1,1,1,1,0,0,0,0]
 => 11110000 => 11100001 => 00011101 => 3
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => 1111000001 => 5
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101011001 => 0111000101 => 3
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101100101 => 1011001001 => 2
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101101001 => 0111001001 => 3
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101110001 => 0011001101 => 2
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0110010101 => 1101010001 => 3
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0110011001 => 0101010101 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0110100101 => 1011010001 => 3
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0110101001 => 0111010001 => 3
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0110110001 => 0011010101 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0111000101 => 1001011001 => 2
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0111001001 => 0101011001 => 2
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0111010001 => 0011011001 => ? ∊ {2,2,3,3,3,3,4}
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0111100001 => 0001011101 => 3
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1001010101 => 1110100001 => 4
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1001011001 => 0110100101 => 2
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1001100101 => 1010101001 => 2
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1001101001 => 0110101001 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001110001 => 0010101101 => 2
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1010010101 => 1101100001 => 4
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1010011001 => 0101100101 => 2
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1010100101 => 1011100001 => 4
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1010101001 => 0111100001 => 4
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1010110001 => 0011100101 => ? ∊ {2,2,3,3,3,3,4}
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1011000101 => 1001101001 => 2
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1011001001 => 0101101001 => 2
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1011010001 => 0011101001 => ? ∊ {2,2,3,3,3,3,4}
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1011100001 => 0001101101 => 3
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1100010101 => 1100110001 => 3
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 1100011001 => 0100110101 => 2
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 1100100101 => 1010110001 => 3
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 1101010001 => 0011110001 => ? ∊ {2,2,3,3,3,3,4}
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1110000101 => 1000111001 => ? ∊ {2,2,3,3,3,3,4}
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 1110001001 => 0100111001 => ? ∊ {2,2,3,3,3,3,4}
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 1110010001 => 0010111001 => ? ∊ {2,2,3,3,3,3,4}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 010101110001 => 001110001101 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => 010110110001 => 001110010101 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => 010111010001 => 001110011001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
 => 101101011000 => 011010110001 => 001110100101 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,1,0,1,1,0,1,0,0,0]
 => 101101101000 => 011011010001 => 001110101001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
 => 101110101000 => 011101010001 => 001110110001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 101111000010 => 011110000101 => 100010111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => 011110001001 => 010010111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,1,1,1,0,0,1,0,0,0]
 => 101111001000 => 011110010001 => 001010111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => 011110100001 => 000110111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,0,0,1,1,1,0,1,0,0,0]
 => 110011101000 => 100111010001 => 001101011001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => 110101011000 => 101010110001 => 001111000101 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => 110101101000 => 101011010001 => 001111001001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => 110110101000 => 101101010001 => 001111010001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,0,1,1,1,0,0,0,0,1,0]
 => 110111000010 => 101110000101 => 100011011001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 110111000100 => 101110001001 => 010011011001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => 110111001000 => 101110010001 => 001011011001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => 110111010000 => 101110100001 => 000111011001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => 111001011000 => 110010110001 => 001101100101 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => 111001101000 => 110011010001 => 001101101001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,1,0,0,0,1,1,0,0]
 => 111010001100 => 110100011001 => 010011100101 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => 111010011000 => 110100110001 => 001011100101 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => 111010101000 => 110101010001 => 001111100001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => 111010110000 => 110101100001 => 000111100101 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,1,1,0,0,0,0,1,0]
 => 111011000010 => 110110000101 => 100011101001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => 111011000100 => 110110001001 => 010011101001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => 111011001000 => 110110010001 => 001011101001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => 111011010000 => 110110100001 => 000111101001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,0,0,1,0,1,0,0,0]
 => 111100101000 => 111001010001 => 001101110001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => 111010000101 => 100011110001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,0,1,0,0,0,1,0,0]
 => 111101000100 => 111010001001 => 010011110001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,0,1,0,0,1,0,0,0]
 => 111101001000 => 111010010001 => 001011110001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,0,1,0,1,0,0,0,0]
 => 111101010000 => 111010100001 => 000111110001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => 111100000101 => 100001111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,1,0,0,0,0,1,0,0]
 => 111110000100 => 111100001001 => 010001111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,1,0,0,0,1,0,0,0]
 => 111110001000 => 111100010001 => 001001111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,1,0,0,1,0,0,0,0]
 => 111110010000 => 111100100001 => 000101111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,1,0,1,0,0,0,0,0]
 => 111110100000 => 111101000001 => 000011111001 => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}

Description

The length of the longest constant subword.

Matching statistic: St001087

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
 => [3,1,2] => [3,2,1] => [1,2,3] => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,4,3,2,1] => [1,2,3,4,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,3,2,1,4] => [1,3,4,5,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,4,2,1,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [5,3,4,2,1] => [1,3,2,4,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,2,1,3,4] => [1,4,5,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,4,3,1,2] => [1,2,3,5,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,3,1,2,4] => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,4,2,3,1] => [1,2,4,3,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,2,5,3,1] => [2,4,1,3,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,3,1,4] => [1,4,3,5,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,4,1,2,3] => [1,2,5,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,3,4,1,2] => [1,3,2,5,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,4,1] => [1,4,3,2,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,4,3,2,1,5] => [1,3,4,5,6,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,5,3,2,1,4] => [1,2,4,5,6,3] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [6,4,5,3,2,1] => [1,3,2,4,5,6] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,3,2,1,4,5] => [1,4,5,6,3,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,5,4,2,1,3] => [1,2,3,5,6,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,4,2,1,3,5] => [1,3,5,6,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,3,6,4,2,1] => [2,4,1,3,5,6] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [6,3,4,2,1,5] => [1,4,3,5,6,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,5,2,1,3,4] => [1,2,5,6,4,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [6,4,5,2,1,3] => [1,3,2,5,6,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [6,3,4,5,2,1] => [1,4,3,2,5,6] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,2,1,3,4,5] => [1,5,6,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,4,3,1,2,5] => [1,3,4,6,5,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,5,3,1,2,4] => [1,2,4,6,5,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [6,4,5,3,1,2] => [1,3,2,4,6,5] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,3,1,2,4,5] => [1,4,6,5,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [6,4,2,3,1,5] => [1,3,5,4,6,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2,6,5,3,1] => [3,5,1,2,4,6] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [2,4,6,1,3,5] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2,6,3,1,5] => [3,5,1,4,6,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [6,5,2,3,1,4] => [1,2,5,4,6,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,2,4,6,3,1] => [2,5,3,1,4,6] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,3,1,4,5] => [1,5,4,6,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [6,4,2,1,7,5,3] => [2,4,6,7,1,3,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,3,7,4,2,1,6] => [3,5,1,4,6,7,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [6,3,5,7,4,2,1] => [2,5,3,1,4,6,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [7,3,4,2,1,5,6] => [1,5,4,6,7,3,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [6,4,7,5,2,1,3] => [2,4,1,3,6,7,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [6,3,4,7,5,2,1] => [2,5,4,1,3,6,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [6,3,7,4,5,2,1] => [2,5,1,4,3,6,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [7,3,4,5,2,1,6] => [1,5,4,3,6,7,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [6,4,7,5,3,1,2] => [2,4,1,3,5,7,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [7,4,2,3,1,5,6] => [1,4,6,5,7,3,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [4,2,7,5,3,1,6] => [4,6,1,3,5,7,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [5,3,1,7,6,4,2] => [3,5,7,1,2,4,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [5,3,1,7,4,2,6] => [3,5,7,1,4,6,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [4,2,7,6,3,1,5] => [4,6,1,2,5,7,3] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [4,2,7,5,6,3,1] => [4,6,1,3,2,5,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [6,3,1,5,7,4,2] => [2,5,7,3,1,4,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [4,2,7,3,1,5,6] => [4,6,1,5,7,3,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [6,4,7,5,2,3,1] => [2,4,1,3,6,5,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [5,2,4,7,6,3,1] => [3,6,4,1,2,5,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [6,3,1,7,4,5,2] => [2,5,7,1,4,3,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [5,2,4,7,3,1,6] => [3,6,4,1,5,7,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [6,2,4,5,7,3,1] => [2,6,4,3,1,5,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,3,7,6,4,1,2] => [3,5,1,2,4,7,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [6,4,1,2,7,5,3] => [2,4,7,6,1,3,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,3,7,4,1,2,6] => [3,5,1,4,7,6,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [6,3,5,7,4,1,2] => [2,5,3,1,4,7,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [5,2,3,7,6,4,1] => [3,6,5,1,2,4,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [6,4,1,7,5,2,3] => [2,4,7,1,3,6,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [5,2,3,7,4,1,6] => [3,6,5,1,4,7,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2,7,6,3,4,1] => [3,6,1,2,5,4,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [6,3,4,1,7,5,2] => [2,5,4,7,1,3,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,7,5,1,2,3,4] => [7,4,1,6,3,5,2] => [1,4,7,2,5,3,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2,7,3,4,1,6] => [3,6,1,5,4,7,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [6,2,3,5,7,4,1] => [2,6,5,3,1,4,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [6,2,5,7,3,4,1] => [2,6,3,1,5,4,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [6,4,7,5,1,2,3] => [2,4,1,3,7,6,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [6,3,4,7,5,1,2] => [2,5,4,1,3,7,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [6,3,7,4,5,1,2] => [2,5,1,4,3,7,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,6,4,5,1,7,3] => [7,3,4,5,1,2,6] => [1,5,4,3,7,6,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [6,2,3,4,7,5,1] => [2,6,5,4,1,3,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [6,2,3,7,4,5,1] => [2,6,5,1,4,3,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [6,2,7,3,4,5,1] => [2,6,1,5,4,3,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,7,4,5,6,1,3] => [7,3,4,5,6,1,2] => [1,5,4,3,2,7,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [7,3,4,5,6,1,2] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5} - 1

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.

Matching statistic: St000877
(load all 3 compositions to match this statistic)

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 10 => 01 => 10 => 0 = 1 - 1
[1,0,1,0]
 => 1010 => 0101 => 1010 => 0 = 1 - 1
[1,1,0,0]
 => 1100 => 0011 => 0101 => 1 = 2 - 1
[1,0,1,0,1,0]
 => 101010 => 010101 => 101010 => 0 = 1 - 1
[1,0,1,1,0,0]
 => 101100 => 010011 => 100101 => 1 = 2 - 1
[1,1,0,0,1,0]
 => 110010 => 001101 => 010110 => 1 = 2 - 1
[1,1,0,1,0,0]
 => 110100 => 001011 => 010101 => 1 = 2 - 1
[1,1,1,0,0,0]
 => 111000 => 000111 => 001011 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 10101010 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 01010011 => 10100101 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 01001101 => 10010110 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 01001011 => 10010101 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 01000111 => 10001011 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 00110101 => 01011010 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 00110011 => 01010101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 00101101 => 01010110 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 00101011 => 01010101 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 00100111 => 01001011 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 00011101 => 00101110 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 00011011 => 00101101 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 00010111 => 00101011 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 00001111 => 00010111 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => 1010101010 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101010011 => 1010100101 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101001101 => 1010010110 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101001011 => 1010010101 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101000111 => 1010001011 => ? ∊ {2,3,3,3,3,4} - 1
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0100110101 => 1001011010 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0100110011 => 1001010101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0100101101 => 1001010110 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0100101011 => 1001010101 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0100100111 => 1001001011 => ? ∊ {2,3,3,3,3,4} - 1
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0100011101 => 1000101110 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0100011011 => 1000101101 => ? ∊ {2,3,3,3,3,4} - 1
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0100010111 => 1000101011 => ? ∊ {2,3,3,3,3,4} - 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0100001111 => 1000010111 => ? ∊ {2,3,3,3,3,4} - 1
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0011010101 => 0101101010 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0011010011 => 0101100101 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0011001101 => 0101010110 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0011001011 => 0101010101 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0011000111 => 0101001011 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 0010110101 => 0101011010 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 0010110011 => 0101010101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 0010101101 => 0101010110 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 0010101011 => 0101010101 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0010100111 => 0101001011 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 0010011101 => 0100101110 => ? ∊ {2,3,3,3,3,4} - 1
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0010011011 => 0100101101 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0010010111 => 0100101011 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0010001111 => 0100010111 => 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0001110101 => 0010111010 => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0001110011 => 0010110101 => 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0001101101 => 0010110110 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0001101011 => 0010110101 => 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0001100111 => 0010101011 => 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 0001011101 => 0010101110 => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => 010101001101 => 101010010110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 010101000111 => 101010001011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => 010100110101 => 101001011010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => 010100101101 => 101001010110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => 010100100111 => 101001001011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => 101011100010 => 010100011101 => 101000101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => 010100011011 => 101000101101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => 010100010111 => 101000101011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => 010100001111 => 101000010111 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 101100101010 => 010011010101 => 100101101010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 101100110010 => 010011001101 => 100101010110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 101100111000 => 010011000111 => 100101001011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => 101101001010 => 010010110101 => 100101011010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => 101101010010 => 010010101101 => 100101010110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => 101101011000 => 010010100111 => 100101001011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => 101101100010 => 010010011101 => 100100101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => 101101100100 => 010010011011 => 100100101101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => 101101101000 => 010010010111 => 100100101011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => 101101110000 => 010010001111 => 100100010111 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => 101110001010 => 010001110101 => 100010111010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 101110001100 => 010001110011 => 100010110101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => 101110010010 => 010001101101 => 100010110110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => 101110010100 => 010001101011 => 100010110101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => 101110011000 => 010001100111 => 100010101011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => 101110100010 => 010001011101 => 100010101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => 010001011011 => 100010101101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => 101110101000 => 010001010111 => 100010101011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => 101110110000 => 010001001111 => 100010010111 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 101111000010 => 010000111101 => 100001011110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => 010000111011 => 100001011101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => 101111001000 => 010000110111 => 100001011011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => 010000101111 => 100001010111 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => 010000011111 => 100000101111 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => 110010111000 => 001101000111 => 010110001011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 110011100010 => 001100011101 => 010100101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => 110101100010 => 001010011101 => 010100101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => 110110001010 => 001001110101 => 010010111010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => 110110010010 => 001001101101 => 010010110110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => 110110100010 => 001001011101 => 010010101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => 110111000010 => 001000111101 => 010001011110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
 => 111000101100 => 000111010011 => 001011100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => 111001100010 => 000110011101 => 001010101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,1,0,1,0,0,0,1,0,1,0]
 => 111010001010 => 000101110101 => 001010111010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => 111010010010 => 000101101101 => 001010110110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1

Description

The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length

$n$ with depth

$k$ is

$\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$ , see [2].

Matching statistic: St000068

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [2,1] => ([],2)
 => 2 = 1 + 1
[1,0,1,0]
 => [3,1,2] => [3,2,1] => ([],3)
 => 3 = 2 + 1
[1,1,0,0]
 => [2,3,1] => [3,1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,3,2,1] => ([],4)
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,3,1,2] => ([(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,3,1] => ([(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,4,3,2,1] => ([],5)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [5,3,4,2,1] => ([(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,4,3,1,2] => ([(3,4)],5)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,4,2,3,1] => ([(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([],6)
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [6,4,5,3,2,1] => ([(4,5)],6)
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(4,5)],6)
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [6,3,4,2,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,5,2,1,3,4] => ([(2,5),(3,5),(5,4)],6)
 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [6,4,5,2,1,3] => ([(1,5),(2,5),(3,4)],6)
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(4,5)],6)
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,4,3,1,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [6,4,5,3,1,2] => ([(2,5),(3,4)],6)
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(4,5)],6)
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [6,4,2,3,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2,6,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,2,4,6,3,1] => ([(1,5),(2,3),(2,4),(4,5)],6)
 => ? ∊ {2,3} + 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [5,2,3,6,4,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
 => ? ∊ {2,3} + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => [7,4,3,2,1,5,6] => ([(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [7,6,3,2,1,4,5] => ([(2,6),(3,6),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [7,3,2,1,4,5,6] => ([(1,6),(2,6),(3,6),(4,5),(6,4)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [7,4,2,1,3,5,6] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [7,6,3,4,2,1,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [6,3,5,7,4,2,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [7,3,4,2,1,5,6] => ([(1,6),(2,6),(3,4),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [7,6,5,2,1,3,4] => ([(3,6),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [7,5,2,1,3,4,6] => ([(1,6),(2,5),(3,5),(4,6),(5,4)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [6,4,7,5,2,1,3] => ([(0,5),(1,5),(2,6),(3,4),(3,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [6,3,4,7,5,2,1] => ([(2,6),(3,4),(4,5),(4,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [6,3,7,4,5,2,1] => ([(2,6),(3,4),(3,6),(4,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [7,6,2,1,3,4,5] => ([(2,6),(3,6),(4,5),(6,4)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [7,4,5,6,2,1,3] => ([(1,6),(2,6),(3,4),(4,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [7,3,4,5,6,2,1] => ([(3,4),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => [7,2,1,3,4,5,6] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [7,6,4,3,1,2,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [7,4,3,1,2,5,6] => ([(1,6),(2,6),(3,4),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => ([(3,6),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [7,6,3,1,2,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [7,6,4,2,3,1,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [7,4,2,3,1,5,6] => ([(1,6),(2,6),(3,4),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [4,2,7,5,6,3,1] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [6,3,1,5,7,4,2] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [4,2,7,3,1,5,6] => ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,6),(6,4)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => [7,6,5,2,3,1,4] => ([(3,6),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => [7,5,2,3,1,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [5,2,4,7,6,3,1] => ([(1,5),(1,6),(2,3),(2,4),(4,5),(4,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [7,4,1,6,2,3,5] => [7,5,2,4,6,3,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [6,3,1,7,4,5,2] => ([(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(5,4)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [5,2,4,7,3,1,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(3,6),(4,5),(4,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => [7,6,2,3,1,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [6,2,4,5,7,3,1] => ([(1,6),(2,3),(2,4),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => [7,6,4,1,2,3,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,3,7,6,4,1,2] => ([(0,5),(0,6),(1,3),(2,4),(2,5),(2,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [6,4,1,2,7,5,3] => ([(0,6),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,3,7,4,1,2,6] => ([(0,5),(0,6),(1,3),(2,4),(2,5),(3,6),(4,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [6,3,5,7,4,1,2] => ([(0,6),(1,4),(2,3),(2,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [5,2,3,7,6,4,1] => ([(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [6,4,1,7,5,2,3] => ([(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [5,2,3,7,4,1,6] => ([(0,6),(1,5),(1,6),(2,3),(3,4),(3,5),(4,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2,7,6,3,4,1] => ([(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [6,3,4,1,7,5,2] => ([(0,6),(1,4),(2,3),(2,5),(2,6),(4,5),(4,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,7,5,1,2,3,4] => [7,4,1,6,3,5,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(4,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2,7,3,4,1,6] => ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => [7,6,2,3,4,1,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [6,2,3,5,7,4,1] => ([(1,3),(2,6),(3,4),(3,5),(5,6)],7)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1

Description

The number of minimal elements in a poset.

Matching statistic: St000144

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [[1],[]]
 => []
 => []
 => ? = 1
[1,0,1,0]
 => [[1,1],[]]
 => []
 => []
 => ? ∊ {1,2}
[1,1,0,0]
 => [[2],[]]
 => []
 => []
 => ? ∊ {1,2}
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [[3],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [[4,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [[4,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [[5,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}

Description

The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form

$1^h0^h$ ) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.

The following 203 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000259The diameter of a connected graph. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001330The hat guessing number of a graph. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000291The number of descents of a binary word. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000381The largest part of an integer composition. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001060The distinguishing index of a graph. St001645The pebbling number of a connected graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000306The bounce count of a Dyck path. St001875The number of simple modules with projective dimension at most 1. St000451The length of the longest pattern of the form k 1 2. St000264The girth of a graph, which is not a tree. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000028The number of stack-sorts needed to sort a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000392The length of the longest run of ones in a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000808The number of up steps of the associated bargraph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000035The number of left outer peaks of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000153The number of adjacent cycles of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000742The number of big ascents of a permutation after prepending zero. St000213The number of weak exceedances (also weak excedences) of a permutation. St000245The number of ascents of a permutation. St000542The number of left-to-right-minima of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000710The number of big deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000702The number of weak deficiencies of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001545The second Elser number of a connected graph. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000474Dyson's crank of a partition. St000993The multiplicity of the largest part of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001432The order dimension of the partition. St001488The number of corners of a skew partition. St000141The maximum drop size of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000477The weight of a partition according to Alladi. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001717The largest size of an interval in a poset. St001720The minimal length of a chain of small intervals in a lattice. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001864The number of excedances of a signed permutation. St001894The depth of a signed permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000983The length of the longest alternating subword. St001896The number of right descents of a signed permutations. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000013The height of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000352The Elizalde-Pak rank of a permutation. St000662The staircase size of the code of a permutation. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000390The number of runs of ones in a binary word. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000097The order of the largest clique of the graph. St001870The number of positive entries followed by a negative entry in a signed permutation. St001893The flag descent of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001589The nesting number of a perfect matching. St001712The number of natural descents of a standard Young tableau. St001960The number of descents of a permutation minus one if its first entry is not one. St000015The number of peaks of a Dyck path. St000317The cycle descent number of a permutation. St000619The number of cyclic descents of a permutation. St000628The balance of a binary word. St000670The reversal length of a permutation. St000765The number of weak records in an integer composition. St000942The number of critical left to right maxima of the parking functions. St000991The number of right-to-left minima of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000646The number of big ascents of a permutation. St000779The tier of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001590The crossing number of a perfect matching. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000166The depth minus 1 of an ordered tree. St000442The maximal area to the right of an up step of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000094The depth of an ordered tree. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001372The length of a longest cyclic run of ones of a binary word. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001624The breadth of a lattice. St000386The number of factors DDU in a Dyck path. St000884The number of isolated descents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001435The number of missing boxes in the first row. St001820The size of the image of the pop stack sorting operator. St001877Number of indecomposable injective modules with projective dimension 2. St000292The number of ascents of a binary word. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation.