- FindStat

Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000259
(load all 3 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001093
(load all 3 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The detour number of a graph. This is the number of vertices in a longest induced path in a graph. Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.

Matching statistic: St000897

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of different multiplicities of parts of an integer partition.

Matching statistic: St000455
(load all 13 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%

Values

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

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: St000353

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner valleys of a permutation. The number of valleys including the boundary is [[St000099]].

Matching statistic: St000711

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of big exceedences of a permutation. A big exceedence of a permutation

$\pi$ is an index

$i$ such that

$\pi(i) - i > 1$ . This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.

Matching statistic: St001212

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.

Matching statistic: St001553
(load all 2 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. The statistic returns zero in case that bimodule is the zero module.

Matching statistic: St000092

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of outer peaks of a permutation. An outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ or

$n$ if

$w_{n} > w_{n-1}$ . In other words, it is a peak in the word

$[0,w_1,..., w_n,0]$ .

Matching statistic: St000264

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%

Values

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

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001487The number of inner corners of a skew partition. St000893The number of distinct diagonal sums of an alternating sign matrix.