- FindStat

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001086
(load all 6 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern

$132$ , where the matched entries are all adjacent.

Matching statistic: St000455

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%

Values

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

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001728

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001728: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of invisible descents of a permutation. A visible descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i+1) \leq \min(i, \pi(i))$ . Thus, an invisible descent satisfies

$\pi(i) > \pi(i+1) > i$ .

Matching statistic: St001208
(load all 4 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ .

Matching statistic: St001729

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of visible descents of a permutation. A visible descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i+1) \leq \min(i, \pi(i))$ .

Matching statistic: St001737

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of descents of type 2 in a permutation. A position

$i\in[1,n-1]$ is a descent of type 2 of a permutation

$\pi$ of

$n$ letters, if it is a descent and if

$\pi(j) < \pi(i)$ for all

$j < i$ .

Matching statistic: St001390

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. For a given permutation

$\pi$ , this is the index of the row containing

$\pi^{-1}(1)$ of the recording tableau of

$\pi$ (obtained by [[Mp00070]]).

Matching statistic: St001960

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].