- FindStat

Your data matches 38 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001091
(load all 4 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.

Matching statistic: St000052
(load all 27 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.

Matching statistic: St000356
(load all 11 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$13\!\!-\!\!2$ .

Matching statistic: St001777

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => []
 => []
 => [0] => ? = 0
[1,0,1,0]
 => [1]
 => [[1]]
 => [1] => 0
[1,1,0,0]
 => []
 => []
 => [0] => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 0
[1,0,1,1,0,0]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[1,1,0,0,1,0]
 => [2]
 => [[1,2]]
 => [2] => 0
[1,1,0,1,0,0]
 => [1]
 => [[1]]
 => [1] => 0
[1,1,1,0,0,0]
 => []
 => []
 => [0] => ? = 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 2
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [[1,2,3]]
 => [3] => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [[1,2]]
 => [2] => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [[1]]
 => [1] => 0
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [0] => ? = 0
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => [1,2,3,3] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => [1,2,2,4] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [1,2,2,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [1,2,2,2] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => [1,1,3,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [1,1,3,3] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [1,1,2,4] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [1,1,2,3] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [1,1,1,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => [2,3,4] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [2,3,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [2,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [2,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [1,3,4] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [1,3,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 2
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 0
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [0] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
 => [1,2,3,4,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
 => [1,2,3,3,5] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
 => [1,2,3,3,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
 => [1,2,3,3,3] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
 => [1,2,2,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
 => [1,2,2,4,4] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
 => [1,2,2,3,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
 => [1,2,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
 => [1,2,2,3,3] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
 => [1,2,2,2,5] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
 => [1,2,2,2,4] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
 => [1,1,3,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
 => [1,1,3,4,4] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
 => [1,1,3,3,5] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
 => [1,1,3,3,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
 => [1,1,3,3,3] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
 => [1,1,2,4,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
 => [1,1,2,4,4] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
 => [1,1,2,3,5] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
 => [1,1,2,3,4] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
 => [1,1,2,2,5] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
 => [1,1,1,4,5] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
 => [1,1,1,4,4] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
 => [1,1,1,3,5] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
 => [2,3,4,5] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
 => [2,3,4,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
 => [2,3,3,5] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2]
 => [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
 => [2,3,3,4] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [2,3,3,3] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
 => [2,2,4,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => [2,2,4,4] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
 => [2,2,3,5] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [[1,2,7,11],[3,4,10],[5,6],[8,9]]
 => [2,2,3,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [[1,2,9,10,11],[3,4],[5,6],[7,8]]
 => [2,2,2,5] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
 => [1,3,4,5] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
 => [1,3,4,4] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
 => [1,3,3,5] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [[1,3,4,11],[2,6,7],[5,9,10],[8]]
 => [1,3,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
 => [1,2,4,5] => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [[1,3,6,7],[2,5,10,11],[4,9],[8]]
 => [1,2,4,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [[1,3,6,10,11],[2,5,9],[4,8],[7]]
 => [1,2,3,5] => ? = 0
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [[1,4,5,6,11],[2,8,9,10],[3],[7]]
 => [1,1,4,5] => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
 => [3,4,5] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,4,3]
 => [[1,2,3,7],[4,5,6,11],[8,9,10]]
 => [3,4,4] => ? = 1

Description

The number of weak descents in an integer composition. A weak descent of an integer composition

$\alpha=(a_1, \dots, a_n)$ is an index

$1\leq i < n$ such that

$a_i \geq a_{i+1}$ .

Matching statistic: St001067

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 70%

Values

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

Description

The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.

Matching statistic: St000932

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].

Matching statistic: St000204
(load all 4 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 60%

Values

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

Description

The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size

$n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an

$(n+1)$ -gon.

Matching statistic: St001167
(load all 7 compositions to match this statistic)

Mapping

Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 60%

Values

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

Description

The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. The top of a module is the cokernel of the inclusion of the radical of the module into the module. For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.

Matching statistic: St001323
(load all 3 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001323: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 60%

Values

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

Description

The independence gap of a graph. This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph. In particular, this statistic is

$0$ for well covered graphs

Matching statistic: St000010

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 90%

Values

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

Description

The length of the partition.

The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St000065The number of entries equal to -1 in an alternating sign matrix. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000358The number of occurrences of the pattern 31-2. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000711The number of big exceedences of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000225Difference between largest and smallest parts in a partition. St000710The number of big deficiencies of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001960The number of descents of a permutation minus one if its first entry is not one. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module