- FindStat

Your data matches 20 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001504
(load all 9 compositions to match this statistic)

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000722

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000722: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of different neighbourhoods in a graph.

Matching statistic: St000831
(load all 6 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 86%

Values

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

Description

The number of indices that are either descents or recoils. This is, for a permutation

$\pi$ of length

$n$ , this statistics counts the set

$\{ 1 \leq i < n : \pi(i) > \pi(i+1) \text{ or } \pi^{-1}(i) > \pi^{-1}(i+1)\}.$

Matching statistic: St000240
(load all 2 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000240: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%

Values

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

Description

The number of indices that are not small excedances. A small excedance is an index

$i$ for which

$\pi_i = i+1$ .

Matching statistic: St001115

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1,0]
 => [(1,2)]
 => [2,1] => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [4,3,2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [10,3,2,5,4,7,6,9,8,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [8,3,2,5,4,7,6,1,10,9] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [6,3,2,5,4,1,10,9,8,7] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [10,3,2,5,4,9,8,7,6,1] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [6,3,2,5,4,1,8,7,10,9] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [4,3,2,1,10,7,6,9,8,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [10,3,2,7,6,5,4,9,8,1] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [10,3,2,9,8,7,6,5,4,1] => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [10,3,2,9,6,5,8,7,4,1] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [4,3,2,1,6,5,8,7,10,9] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,10,5,4,7,6,9,8,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,8,5,4,7,6,3,10,9] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,6,5,4,3,8,7,10,9] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [10,5,4,3,2,7,6,9,8,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [10,7,6,5,4,3,2,9,8,1] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [10,9,8,5,4,7,6,3,2,1] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [10,5,4,3,2,9,8,7,6,1] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [10,9,6,5,4,3,8,7,2,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [6,5,4,3,2,1,8,7,10,9] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [14,3,2,5,4,7,6,9,8,11,10,13,12,1] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11),(13,14)]
 => [12,3,2,5,4,7,6,9,8,11,10,1,14,13] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9),(11,14),(12,13)]
 => [10,3,2,5,4,7,6,9,8,1,14,13,12,11] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,13),(11,12)]
 => [14,3,2,5,4,7,6,9,8,13,12,11,10,1] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12),(13,14)]
 => [10,3,2,5,4,7,6,9,8,1,12,11,14,13] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,14),(10,11),(12,13)]
 => [8,3,2,5,4,7,6,1,14,11,10,13,12,9] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11),(13,14)]
 => [8,3,2,5,4,7,6,1,12,11,10,9,14,13] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,11),(9,10),(12,13)]
 => [14,3,2,5,4,7,6,11,10,9,8,13,12,1] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
 => [14,3,2,5,4,7,6,13,12,11,10,9,8,1] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10),(13,14)]
 => [12,3,2,5,4,7,6,11,10,9,8,1,14,13] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10),(11,14),(12,13)]
 => [8,3,2,5,4,7,6,1,10,9,14,13,12,11] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,14),(10,13),(11,12)]
 => [8,3,2,5,4,7,6,1,14,13,12,11,10,9] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,10),(11,12)]
 => [14,3,2,5,4,7,6,13,10,9,12,11,8,1] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12),(13,14)]
 => [8,3,2,5,4,7,6,1,10,9,12,11,14,13] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,9),(10,11),(12,13)]
 => [6,3,2,5,4,1,14,9,8,11,10,13,12,7] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11),(13,14)]
 => [6,3,2,5,4,1,12,9,8,11,10,7,14,13] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9),(11,14),(12,13)]
 => [6,3,2,5,4,1,10,9,8,7,14,13,12,11] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,9),(10,13),(11,12)]
 => [6,3,2,5,4,1,14,9,8,13,12,11,10,7] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12),(13,14)]
 => [6,3,2,5,4,1,10,9,8,7,12,11,14,13] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,9),(7,8),(10,11),(12,13)]
 => [14,3,2,5,4,9,8,7,6,11,10,13,12,1] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11),(13,14)]
 => [12,3,2,5,4,9,8,7,6,11,10,1,14,13] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
 => [14,3,2,5,4,11,10,9,8,7,6,13,12,1] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,9),(10,11)]
 => [14,3,2,5,4,13,12,9,8,11,10,7,6,1] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
 => [12,3,2,5,4,11,10,9,8,7,6,1,14,13] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8),(11,14),(12,13)]
 => [10,3,2,5,4,9,8,7,6,1,14,13,12,11] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,9),(7,8),(10,13),(11,12)]
 => [14,3,2,5,4,9,8,7,6,13,12,11,10,1] => ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
 => [14,3,2,5,4,13,10,9,8,7,12,11,6,1] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12),(13,14)]
 => [10,3,2,5,4,9,8,7,6,1,12,11,14,13] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,14),(10,11),(12,13)]
 => [6,3,2,5,4,1,8,7,14,11,10,13,12,9] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11),(13,14)]
 => [6,3,2,5,4,1,8,7,12,11,10,9,14,13] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,11),(9,10),(12,13)]
 => [6,3,2,5,4,1,14,11,10,9,8,13,12,7] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
 => [6,3,2,5,4,1,14,13,12,11,10,9,8,7] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10),(13,14)]
 => [6,3,2,5,4,1,12,11,10,9,8,7,14,13] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,11),(7,8),(9,10),(12,13)]
 => [14,3,2,5,4,11,8,7,10,9,6,13,12,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
 => [14,3,2,5,4,13,8,7,12,11,10,9,6,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => [14,3,2,5,4,13,12,11,10,9,8,7,6,1] => ? = 7 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10),(13,14)]
 => [12,3,2,5,4,11,8,7,10,9,6,1,14,13] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,14),(12,13)]
 => [6,3,2,5,4,1,8,7,10,9,14,13,12,11] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,14),(10,13),(11,12)]
 => [6,3,2,5,4,1,8,7,14,13,12,11,10,9] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,10),(11,12)]
 => [6,3,2,5,4,1,14,13,10,9,12,11,8,7] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,8),(9,10),(11,12)]
 => [14,3,2,5,4,13,8,7,10,9,12,11,6,1] => ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12),(13,14)]
 => [6,3,2,5,4,1,8,7,10,9,12,11,14,13] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [4,3,2,1,14,7,6,9,8,11,10,13,12,5] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11),(13,14)]
 => [4,3,2,1,12,7,6,9,8,11,10,5,14,13] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,14),(12,13)]
 => [4,3,2,1,10,7,6,9,8,5,14,13,12,11] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,14),(6,7),(8,9),(10,13),(11,12)]
 => [4,3,2,1,14,7,6,9,8,13,12,11,10,5] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12),(13,14)]
 => [4,3,2,1,10,7,6,9,8,5,12,11,14,13] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,14),(10,11),(12,13)]
 => [4,3,2,1,8,7,6,5,14,11,10,13,12,9] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14)]
 => [4,3,2,1,8,7,6,5,12,11,10,9,14,13] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,14),(6,7),(8,11),(9,10),(12,13)]
 => [4,3,2,1,14,7,6,11,10,9,8,13,12,5] => ? = 6 - 1

Description

The number of even descents of a permutation.

Matching statistic: St001645

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%

Values

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

Description

The pebbling number of a connected graph.

Matching statistic: St001182
(load all 2 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%

Values

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

Description

Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.

Matching statistic: St000672
(load all 2 compositions to match this statistic)

Mapping

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [5,4,1,6,3,2] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [2,1,7,6,3,8,5,4] => ? = 4 - 1
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [5,4,1,6,3,2,8,7] => ? = 4 - 1
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => ? = 4 - 1
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [7,6,5,4,1,8,3,2] => ? = 3 - 1
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => ? = 3 - 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => ? = 3 - 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,9,8,5,10,7,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,8,5,4,7,6,3,10,9] => [2,1,7,6,3,8,5,4,10,9] => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,10,5,4,7,6,9,8,3] => [2,1,9,8,3,7,6,10,5,4] => ? = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => [2,1,9,8,7,6,3,10,5,4] => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,8,7,6,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => [2,1,9,8,3,10,7,6,5,4] => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,10,9,6,5,8,7,4,3] => [2,1,8,7,4,3,10,9,6,5] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [4,3,2,1,10,7,6,9,8,5] => [4,3,2,1,9,8,5,10,7,6] => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [6,3,2,5,4,1,8,7,10,9] => [5,4,1,6,3,2,8,7,10,9] => ? = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [6,3,2,5,4,1,10,9,8,7] => [5,4,1,6,3,2,10,9,8,7] => ? = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [8,3,2,5,4,7,6,1,10,9] => [7,6,1,5,4,8,3,2,10,9] => ? = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [10,3,2,5,4,7,6,9,8,1] => [9,8,1,5,4,7,6,10,3,2] => ? = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [10,3,2,5,4,9,8,7,6,1] => [9,8,7,6,1,5,4,10,3,2] => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => [7,6,5,4,1,8,3,2,10,9] => ? = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [10,3,2,7,6,5,4,9,8,1] => [9,8,1,7,6,5,4,10,3,2] => ? = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [10,3,2,9,6,5,8,7,4,1] => [8,7,4,1,9,6,5,10,3,2] => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [10,3,2,9,8,7,6,5,4,1] => [9,8,7,6,5,4,1,10,3,2] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,10,9,8,7] => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => [7,6,1,8,5,4,3,2,10,9] => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [10,5,4,3,2,7,6,9,8,1] => [9,8,1,7,6,10,5,4,3,2] => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [10,5,4,3,2,9,8,7,6,1] => [9,8,7,6,1,10,5,4,3,2] => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [8,7,4,3,6,5,2,1,10,9] => [6,5,2,1,8,7,4,3,10,9] => ? = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [10,7,4,3,6,5,2,9,8,1] => [9,8,1,6,5,2,10,7,4,3] => ? = 4 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [10,9,4,3,6,5,8,7,2,1] => [8,7,2,1,6,5,10,9,4,3] => ? = 5 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [10,9,4,3,8,7,6,5,2,1] => [8,7,6,5,2,1,10,9,4,3] => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [10,7,6,5,4,3,2,9,8,1] => [9,8,1,10,7,6,5,4,3,2] => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [10,9,6,5,4,3,8,7,2,1] => [8,7,2,1,10,9,6,5,4,3] => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [10,9,8,5,4,7,6,3,2,1] => [7,6,3,2,1,10,9,8,5,4] => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,12,11,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
 => [2,1,4,3,6,5,10,9,8,7,12,11] => [2,1,4,3,6,5,10,9,8,7,12,11] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,12,9,8,11,10,7] => [2,1,4,3,6,5,11,10,7,12,9,8] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,12,11,10,9,8,7] => [2,1,4,3,6,5,12,11,10,9,8,7] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
 => [2,1,4,3,8,7,6,5,10,9,12,11] => [2,1,4,3,8,7,6,5,10,9,12,11] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
 => [2,1,4,3,8,7,6,5,12,11,10,9] => [2,1,4,3,8,7,6,5,12,11,10,9] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
 => [2,1,4,3,10,7,6,9,8,5,12,11] => [2,1,4,3,9,8,5,10,7,6,12,11] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,12,7,6,9,8,11,10,5] => [2,1,4,3,11,10,5,9,8,12,7,6] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
 => [2,1,4,3,12,7,6,11,10,9,8,5] => [2,1,4,3,11,10,9,8,5,12,7,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
 => [2,1,4,3,10,9,8,7,6,5,12,11] => [2,1,4,3,10,9,8,7,6,5,12,11] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,12,9,8,7,6,11,10,5] => [2,1,4,3,11,10,5,12,9,8,7,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,12,11,8,7,10,9,6,5] => [2,1,4,3,10,9,6,5,12,11,8,7] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,12,11,10,9,8,7,6,5] => [2,1,4,3,12,11,10,9,8,7,6,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
 => [2,1,6,5,4,3,8,7,10,9,12,11] => [2,1,6,5,4,3,8,7,10,9,12,11] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
 => [2,1,6,5,4,3,8,7,12,11,10,9] => [2,1,6,5,4,3,8,7,12,11,10,9] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
 => [2,1,6,5,4,3,10,9,8,7,12,11] => [2,1,6,5,4,3,10,9,8,7,12,11] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
 => [2,1,6,5,4,3,12,9,8,11,10,7] => [2,1,6,5,4,3,11,10,7,12,9,8] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
 => [2,1,6,5,4,3,12,11,10,9,8,7] => [2,1,6,5,4,3,12,11,10,9,8,7] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,8,5,4,7,6,3,10,9,12,11] => [2,1,7,6,3,8,5,4,10,9,12,11] => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,8,5,4,7,6,3,12,11,10,9] => [2,1,7,6,3,8,5,4,12,11,10,9] => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,10,5,4,7,6,9,8,3,12,11] => [2,1,9,8,3,7,6,10,5,4,12,11] => ? = 6 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,12,5,4,7,6,9,8,11,10,3] => [2,1,11,10,3,7,6,9,8,12,5,4] => ? = 6 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,12,5,4,7,6,11,10,9,8,3] => [2,1,11,10,9,8,3,7,6,12,5,4] => ? = 5 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,10,5,4,9,8,7,6,3,12,11] => [2,1,9,8,7,6,3,10,5,4,12,11] => ? = 5 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,12,5,4,9,8,7,6,11,10,3] => [2,1,11,10,3,9,8,7,6,12,5,4] => ? = 6 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,12,5,4,11,8,7,10,9,6,3] => [2,1,10,9,6,3,11,8,7,12,5,4] => ? = 5 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,12,5,4,11,10,9,8,7,6,3] => [2,1,11,10,9,8,7,6,3,12,5,4] => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
 => [2,1,8,7,6,5,4,3,10,9,12,11] => [2,1,8,7,6,5,4,3,10,9,12,11] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
 => [2,1,8,7,6,5,4,3,12,11,10,9] => [2,1,8,7,6,5,4,3,12,11,10,9] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
 => [2,1,10,7,6,5,4,9,8,3,12,11] => [2,1,9,8,3,10,7,6,5,4,12,11] => ? = 5 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,12,7,6,5,4,9,8,11,10,3] => [2,1,11,10,3,9,8,12,7,6,5,4] => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
 => [2,1,10,9,8,7,6,5,4,3,12,11] => [2,1,10,9,8,7,6,5,4,3,12,11] => 2 = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,12,11,10,9,8,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [4,3,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [4,3,2,1,6,5,8,7,12,11,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 3 = 4 - 1

Description

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

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

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

Matching statistic: St001769

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%

Values

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

Description

The reflection length of a signed permutation. This is the minimal numbers of reflections needed to express a signed permutation.

Matching statistic: St001596
(load all 4 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%

Values

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

Description

The number of two-by-two squares inside a skew partition. This is, the number of cells

$(i,j)$ in a skew partition for which the box

$(i+1,j+1)$ is also a cell inside the skew partition.

The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000670The reversal length of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001668The number of points of the poset minus the width of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path.