Processing math: 100%

Your data matches 16 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001484
Mapping
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 1
[1,1]
 => 0
[3]
 => 1
[2,1]
 => 2
[1,1,1]
 => 0
[4]
 => 1
[3,1]
 => 2
[2,2]
 => 0
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 1
[4,1]
 => 2
[3,2]
 => 2
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 1
[5,1]
 => 2
[4,2]
 => 2
[4,1,1]
 => 1
[3,3]
 => 0
[3,2,1]
 => 3
[3,1,1,1]
 => 1
[2,2,2]
 => 0
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
[7]
 => 1
[6,1]
 => 2
[5,2]
 => 2
[5,1,1]
 => 1
[4,3]
 => 2
[4,2,1]
 => 3
[4,1,1,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 0
[8]
 => 1
[7,1]
 => 2
[6,2]
 => 2
[6,1,1]
 => 1
[5,3]
 => 2
[5,2,1]
 => 3
Description
The number of singletons of an integer partition. A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000932
(load all 8 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 73%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ?
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ?
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ?
 => ? = 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ?
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ?
 => ? = 1
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => ?
 => ? = 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [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]
 => ? = 0
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ?
 => ? = 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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ?
 => ? = 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 0
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ?
 => ? = 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? = 2
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ? = 2
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ?
 => ? = 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ? = 2
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ?
 => ? = 3
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ?
 => ? = 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => ?
 => ? = 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 0
[3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ?
 => ? = 2
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => ?
 => ? = 1
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ?
 => ? = 0
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ?
 => ? = 0
[2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ?
 => ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 0
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ?
 => ? = 1
[11,1]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? = 2
[10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ? = 2
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ?
 => ? = 1
[9,3]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ? = 2
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ?
 => ? = 3
[9,1,1,1]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ?
 => ? = 1
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ?
 => ? = 2
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ?
 => ? = 3
[8,2,2]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ?
 => ? = 1
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: St000502
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 73%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => {{1,2}}
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7}}
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2},{3},{4}}
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,7},{2},{3},{4},{5},{6}}
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7},{8}}
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5},{6,7}}
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => {{1,2},{3},{4,6},{5}}
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => {{1,6},{2},{3},{4,5}}
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => {{1,4,6},{2},{3},{5}}
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,6,7},{2},{3},{4},{5}}
 => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7,8}}
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4},{5,6},{7}}
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => {{1,2},{3},{4},{5,7},{6}}
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => {{1,2},{3,6},{4},{5}}
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => {{1,3},{2},{4},{5},{6}}
 => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,7,8},{2},{3},{4},{5},{6}}
 => ? = 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7},{8,9}}
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4},{5},{6,7},{8}}
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1,2},{3},{4},{5},{6,8},{7}}
 => ? = 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => {{1,8},{2},{3},{4},{5},{6,7}}
 => ? = 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => {{1,6,8},{2},{3},{4},{5},{7}}
 => ? = 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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,8,9},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7},{8},{9,10}}
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7,8},{9}}
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7,9},{8}}
 => ? = 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4},{5,6},{7},{8}}
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4},{5},{6,7,8}}
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1,2},{3},{4},{5,8},{6},{7}}
 => ? = 1
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => {{1,8},{2},{3},{4},{5,6},{7}}
 => ? = 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => {{1,6,7,8},{2},{3},{4},{5}}
 => ? = 2
[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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => {{1,9},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => {{1,5,8},{2},{3},{4},{6},{7}}
 => ? = 0
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => {{1,7,9},{2},{3},{4},{5},{6},{8}}
 => ? = 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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,9,10},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => ? = 0
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11},{12}}
 => ? = 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
 => ? = 2
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7},{8,9},{10}}
 => ? = 2
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7},{8,10},{9}}
 => ? = 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6,7},{8},{9}}
 => ? = 2
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4},{5},{6},{7,8,9}}
 => ? = 3
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1,2},{3},{4},{5},{6,9},{7},{8}}
 => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4,5},{6},{7},{8}}
 => ? = 2
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6},{7,8}}
 => ? = 3
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => {{1,2},{3},{4},{5,7},{6},{8}}
 => ? = 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => {{1,2},{3},{4},{5,7,8},{6}}
 => ? = 2
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1,2},{3},{4,8},{5},{6},{7}}
 => ? = 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,8},{2},{3},{4,5},{6},{7}}
 => ? = 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => {{1,7,8},{2},{3},{4},{5,6}}
 => ? = 2
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => {{1,9},{2},{3},{4},{5},{6,7},{8}}
 => ? = 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => {{1,8},{2},{3},{4},{5,7},{6}}
 => ? = 0
[3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => {{1,5,7,8},{2},{3},{4},{6}}
 => ? = 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => {{1,7,8,9},{2},{3},{4},{5},{6}}
 => ? = 2
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => {{1,10},{2},{3},{4},{5},{6},{7},{8,9}}
 => ? = 1
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => {{1,4,8},{2},{3},{5},{6},{7}}
 => ? = 0
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => {{1,6,9},{2},{3},{4},{5},{7},{8}}
 => ? = 0
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => {{1,8,10},{2},{3},{4},{5},{6},{7},{9}}
 => ? = 0
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: St001067
(load all 8 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 64%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 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]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 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]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 1
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[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]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => ? = 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]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ? = 2
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? = 2
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? = 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 3
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 3
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? = 2
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => ? = 0
[3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 2
[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]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 0
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => ? = 0
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001189
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 64%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 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]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [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
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [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
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 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]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [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
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [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
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [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
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 2
[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]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 0
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 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]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 2
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 0
[3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 2
[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]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 0
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 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: St000441
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 91%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,2] => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,1,7] => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,1,2] => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,1,2] => ? = 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,1,2] => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,1,6] => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,1,7] => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,1,8] => 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3,1,2] => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,1,2] => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,1,2] => ? = 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,3,1,2] => ? = 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,1,2] => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,1,2] => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,3] => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 3
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,2,1,7] => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,9,7,6,5,4,3,1,2] => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,6,8,5,4,3,1,2] => ? = 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,1,2] => ? = 2
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,1,2] => ? = 3
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,3,1,2] => ? = 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [7,5,6,4,3,2,1,8] => ? = 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [11,10,9,8,7,6,5,4,3,1,2] => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [9,10,8,7,6,5,4,3,1,2] => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [9,7,8,6,5,4,3,1,2] => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [8,7,9,6,5,4,3,1,2] => ? = 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,1,2] => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,6,5,8,4,3,1,2] => ? = 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,1,2] => ? = 3
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,1,2] => ? = 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,1,2] => ? = 2
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,7,1,2] => ? = 1
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,3] => ? = 0
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [6,5,4,2,3,1,7] => ? = 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,2,1,7] => ? = 2
[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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [8,6,7,5,4,3,2,1,9] => ? = 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,1,6] => ? = 0
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [12,11,10,9,8,7,6,5,4,3,1,2] => ? = 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [10,11,9,8,7,6,5,4,3,1,2] => ? = 2
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [10,8,9,7,6,5,4,3,1,2] => ? = 2
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [9,8,10,7,6,5,4,3,1,2] => ? = 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [9,8,6,7,5,4,3,1,2] => ? = 2
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [7,8,9,6,5,4,3,1,2] => ? = 3
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,7,6,9,5,4,3,1,2] => ? = 1
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,3,1,2] => ? = 3
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [8,6,5,7,4,3,1,2] => ? = 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,3,1,2] => ? = 2
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,8,3,1,2] => ? = 1
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,1,2] => ? = 3
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,1,2] => ? = 3
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,3,1,2] => ? = 2
[6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,1,2] => ? = 2
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,1,2] => ? = 2
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,1,2,7] => ? = 1
[5,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,1,3] => ? = 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [5,6,4,2,3,1,7] => ? = 2
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [7,6,5,3,4,2,1,8] => ? = 1
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,2,1,7] => ? = 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,3,2,1,8] => ? = 2
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [8,7,5,6,4,3,2,1,9] => ? = 1
[3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,1,7] => ? = 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [7,5,4,6,3,2,1,8] => ? = 0
Description
The number of successions of a permutation. A succession of a permutation π is an index i such that π(i)+1=π(i+1). Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St001640
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 64%
Values
[1]
 => [1,0,1,0]
 => [1,2] => [1,2] => 1
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,3,1,4] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,3,4,2] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [2,3,4,1,5] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,3,4,5,2] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => [2,3,4,5,1,6] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [3,4,1,2,5] => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,5,2,3] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,2,3,4,5] => [1,3,4,5,6,2] => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,1,2,3,6] => [3,4,5,1,2,6] => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [2,4,1,3,5] => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [3,1,4,2,5] => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,3,1,5,4] => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,3,5,2,4] => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,4,5,3] => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,4,2,5,3] => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,2,3,4] => [1,4,5,6,2,3] => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,2,3,4,5,6] => [1,3,4,5,6,7,2] => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6,8] => [2,3,4,5,6,7,1,8] => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,1,5,2,3,6] => [2,4,5,1,3,6] => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,1,2,4,6] => [3,4,1,5,2,6] => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,3,4,2,5] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,6,3,4] => [1,3,5,6,2,4] => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,4,5,3] => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,2,3,5] => [1,4,5,2,6,3] => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,2,3,4,5] => [1,4,5,6,7,2,3] => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7,9] => [2,3,4,5,6,7,8,1,9] => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [6,7,1,2,3,4,5,8] => [3,4,5,6,7,1,2,8] => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [5,1,6,2,3,4,7] => [2,4,5,6,1,3,7] => ? = 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,1,2,5,3,6] => [2,3,5,1,4,6] => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,1,2,6] => [4,5,1,2,3,6] => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,1,3,4,6] => [3,1,4,5,2,6] => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,1,2,3,6,5] => [2,3,4,1,6,5] => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => 2
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,4] => [1,3,4,6,2,5] => 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,7,8,2,3,4,5,6] => [1,4,5,6,7,8,2,3] => ? = 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9,2,3,4,5,6,7,8] => [1,3,4,5,6,7,8,9,2] => ? = 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8,10] => [2,3,4,5,6,7,8,9,1,10] => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [7,8,1,2,3,4,5,6,9] => [3,4,5,6,7,8,1,2,9] => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [6,1,7,2,3,4,5,8] => [2,4,5,6,7,1,3,8] => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [5,7,1,2,3,4,6,8] => [3,4,5,6,1,7,2,8] => ? = 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [5,1,2,6,3,4,7] => [2,3,5,6,1,4,7] => ? = 2
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 3
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [3,6,1,2,4,5,7] => [3,4,1,5,6,2,7] => ? = 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,7,2,8,3,4,5,6] => [1,3,5,6,7,8,2,4] => ? = 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,6,8,2,3,4,5,7] => [1,4,5,6,7,2,8,3] => ? = 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]
 => [1,8,9,2,3,4,5,6,7] => [1,4,5,6,7,8,9,2,3] => ? = 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,10,2,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,10,2] => ? = 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9,11] => [2,3,4,5,6,7,8,9,10,1,11] => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [8,9,1,2,3,4,5,6,7,10] => [3,4,5,6,7,8,9,1,2,10] => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [7,1,8,2,3,4,5,6,9] => [2,4,5,6,7,8,1,3,9] => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [6,8,1,2,3,4,5,7,9] => [3,4,5,6,7,1,8,2,9] => ? = 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [6,1,2,7,3,4,5,8] => [2,3,5,6,7,1,4,8] => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [5,6,7,1,2,3,4,8] => [4,5,6,7,1,2,3,8] => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [4,7,1,2,3,5,6,8] => [3,4,5,1,6,7,2,8] => ? = 1
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [5,1,2,3,6,4,7] => [2,3,4,6,1,5,7] => ? = 2
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [4,5,1,6,2,3,7] => [3,5,6,1,2,4,7] => ? = 3
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [4,1,6,2,3,5,7] => [2,4,5,1,6,3,7] => ? = 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [3,5,6,1,2,4,7] => [4,5,1,6,2,3,7] => ? = 2
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 1
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 0
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,5,6,2,7,3,4] => [1,4,6,7,2,3,5] => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,7,2,3,8,4,5,6] => [1,3,4,6,7,8,2,5] => ? = 1
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? = 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,6,7,8,2,3,4,5] => [1,5,6,7,8,2,3,4] => ? = 2
[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]
 => [1,8,2,9,3,4,5,6,7] => [1,3,5,6,7,8,9,2,4] => ? = 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,3,4,5,6] => [2,1,4,5,6,7,3] => ? = 0
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,5,8,2,3,4,6,7] => [1,4,5,6,2,7,8,3] => ? = 0
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,7,9,2,3,4,5,6,8] => [1,4,5,6,7,8,2,9,3] => ? = 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]
 => [1,9,10,2,3,4,5,6,7,8] => [1,4,5,6,7,8,9,10,2,3] => ? = 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,11,2,3,4,5,6,7,8,9,10] => [1,3,4,5,6,7,8,9,10,11,2] => ? = 0
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10,12] => [2,3,4,5,6,7,8,9,10,11,1,12] => ? = 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,10,1,2,3,4,5,6,7,8,11] => ? => ? = 2
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,9,2,3,4,5,6,7,10] => ? => ? = 2
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [7,9,1,2,3,4,5,6,8,10] => [3,4,5,6,7,8,1,9,2,10] => ? = 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [7,1,2,8,3,4,5,6,9] => ? => ? = 2
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000214
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 91%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,1,4,3] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [2,1,3,5,4] => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [2,1,3,4,6,5] => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,3,2,5,4] => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,1,5,4] => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,4,2,5,3] => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,2,4,5,6,1] => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [2,1,3,4,5,7,6] => ? = 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [1,3,2,4,6,5] => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [2,3,1,4,6,5] => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,1,5,4] => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,2,1,5,3] => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,1,5,2] => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [2,4,3,5,6,1] => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,4,2,5,6,1] => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,8,1] => 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,8,7] => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [1,3,2,4,5,7,6] => ? = 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [2,3,1,4,5,7,6] => ? = 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,2,4,3,6,5] => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [3,2,1,4,6,5] => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [2,3,4,1,6,5] => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [1,2,5,3,6,4] => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,1,5,4,3] => 3
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => [2,4,3,5,6,7,1] => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,9,8] => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => [1,3,2,4,5,6,8,7] => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => [2,3,1,4,5,6,8,7] => ? = 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => [1,2,4,3,5,7,6] => ? = 2
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => [3,2,1,4,5,7,6] => ? = 3
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,7,1] => [2,3,5,4,6,7,1] => ? = 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => [2,4,3,5,6,7,8,1] => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,10,9] => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => [1,3,2,4,5,6,7,9,8] => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => [2,3,1,4,5,6,7,9,8] => ? = 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => [1,2,4,3,5,6,8,7] => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,8,7] => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => [2,3,4,1,5,6,8,7] => ? = 1
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [7,5,1,2,3,4,6] => [1,2,3,5,4,7,6] => ? = 2
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [7,4,2,1,3,5,6] => [2,1,4,3,5,7,6] => ? = 3
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [7,3,4,1,2,5,6] => [3,1,4,2,5,7,6] => ? = 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,1,5,6] => [3,2,4,1,5,7,6] => ? = 2
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [6,2,3,4,5,7,1] => [2,3,4,6,5,7,1] => ? = 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,6,7,1] => [3,2,5,4,6,7,1] => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => [2,3,5,4,6,7,8,1] => ? = 1
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,7,1] => [4,2,5,3,6,7,1] => ? = 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]
 => [4,2,3,5,6,7,8,9,1] => [2,4,3,5,6,7,8,9,1] => ? = 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [3,4,5,6,1,7,2] => ? = 0
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,12,11] => ? = 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,11,10] => ? = 2
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,1,2,4,5,6,7,8,9] => [1,3,2,4,5,6,7,8,10,9] => ? = 2
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,3,1,4,5,6,7,8,9] => [2,3,1,4,5,6,7,8,10,9] => ? = 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [9,4,1,2,3,5,6,7,8] => [1,2,4,3,5,6,7,9,8] => ? = 2
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,9,8] => ? = 3
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,4,1,5,6,7,8] => [2,3,4,1,5,6,7,9,8] => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [8,5,1,2,3,4,6,7] => [1,2,3,5,4,6,8,7] => ? = 2
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [8,4,2,1,3,5,6,7] => [2,1,4,3,5,6,8,7] => ? = 3
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [8,3,4,1,2,5,6,7] => [3,1,4,2,5,6,8,7] => ? = 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => [3,2,4,1,5,6,8,7] => ? = 2
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => [2,3,4,5,1,6,8,7] => ? = 1
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => ? = 2
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [7,5,2,1,3,4,6] => [2,1,3,5,4,7,6] => ? = 3
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => [1,4,3,2,5,7,6] => ? = 3
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [7,4,2,3,1,5,6] => [2,4,3,1,5,7,6] => ? = 2
[6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [7,3,4,2,1,5,6] => [3,4,2,1,5,7,6] => ? = 2
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,5,1,6] => [3,2,4,5,1,7,6] => ? = 2
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => [2,3,4,5,7,6,1] => ? = 1
[5,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [6,7,2,1,3,4,5] => [2,1,3,6,4,7,5] => ? = 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => [3,2,4,6,5,7,1] => ? = 2
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => [2,3,4,6,5,7,8,1] => ? = 1
[4,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,7,1] => [2,5,4,3,6,7,1] => ? = 2
Description
The number of adjacencies of a permutation. An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000445
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 82%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0]
 => ? = 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0,0]
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0]
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,0]
 => ? = 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => 2
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 1
[2,2,1,1,1,1]
 => [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]
 => [1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 0
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0]
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0,0]
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0,0]
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0,0]
 => ? = 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0]
 => ? = 2
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0,0]
 => ? = 3
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1
[4,1,1,1,1,1]
 => [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]
 => [1,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? = 1
[2,2,2,1,1,1]
 => [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]
 => [1,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? = 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]
 => [1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0]
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0,0,0]
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0,0,0]
 => ? = 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,0]
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0,0]
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => ? = 1
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0]
 => ? = 2
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
 => ? = 3
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0,0]
 => ? = 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0]
 => ? = 2
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0]
 => ? = 0
[5,1,1,1,1,1]
 => [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]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => ? = 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => ? = 1
[3,3,1,1,1,1]
 => [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]
 => [1,1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 0
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? = 2
[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]
 => [1,1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => ? = 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 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]
 => [1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St001223
(load all 8 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 55%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 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]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 2
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => ? = 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 0
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 3
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => ? = 1
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => ? = 0
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 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]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 1
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 3
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 2
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 0
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => ? = 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[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]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 0
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.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001948The number of augmented double ascents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.