- FindStat

Your data matches 32 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000445
(load all 12 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of rises of length 1 of a Dyck path.

Matching statistic: St000932
(load all 101 compositions to match this statistic)

Mapping

Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 86%●distinct values known / distinct values provided: 80%

Values

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

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: St001067
(load all 55 compositions to match this statistic)

Mapping

Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 81%●distinct values known / distinct values provided: 70%

Values

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

Description

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

Matching statistic: St001189
(load all 15 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 81%●distinct values known / distinct values provided: 70%

Values

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

Description

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000502
(load all 11 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of successions of a set partitions. This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.

Matching statistic: St000504

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 80%

Values

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

Description

The cardinality of the first block of a set partition. The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].

Matching statistic: St000475

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000441
(load all 9 compositions to match this statistic)

Mapping

Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of successions of a permutation. A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.

Matching statistic: St001223
(load all 55 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 60%

Values

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

Description

Number 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.

Matching statistic: St001233
(load all 23 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 60%

Values

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

Description

The number of indecomposable 2-dimensional modules with projective dimension one.

The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000931The number of occurrences of the pattern UUU in a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000247The number of singleton blocks of a set partition. St000925The number of topologically connected components of a set partition. St000248The number of anti-singletons of a set partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001948The number of augmented double ascents of a permutation. St000022The number of fixed points of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000221The number of strong fixed points of a permutation. St000732The number of double deficiencies of a permutation. St000989The number of final rises of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001621The number of atoms of a lattice.