searching the database
Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> 1
[1,0,1,0]
=> [1]
=> 2
[1,1,0,0]
=> []
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> 5
[1,0,1,1,0,0]
=> [1,1]
=> 3
[1,1,0,0,1,0]
=> [2]
=> 3
[1,1,0,1,0,0]
=> [1]
=> 2
[1,1,1,0,0,0]
=> []
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 14
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 9
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 10
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 4
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 9
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 6
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 7
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3]
=> 4
[1,1,1,0,0,1,0,0]
=> [2]
=> 3
[1,1,1,0,1,0,0,0]
=> [1]
=> 2
[1,1,1,1,0,0,0,0]
=> []
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 42
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 28
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 32
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 23
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 32
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 22
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 26
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 19
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 12
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 17
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 28
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 19
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 22
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 16
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 23
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 16
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 19
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 14
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 4
Description
The number of partitions contained in the given partition.
Matching statistic: St000420
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 97%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 97%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 14
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 9
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 9
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 7
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 42
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 28
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 32
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 23
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 32
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 22
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 26
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 19
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 12
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 17
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 28
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 19
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 22
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 16
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 23
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 16
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 19
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 14
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 14
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 10
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 12
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 9
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 26
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 34
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 21
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 29
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 22
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 15
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9
[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]
=> [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]
=> ? = 9
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 10
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 17
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 17
[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]
=> [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]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 19
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 24
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 25
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 25
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [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]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 24
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 19
[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]
=> [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]
=> ? = 11
[]
=> []
=> []
=> []
=> ? = 1
[1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 26
[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 34
[1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 21
[1,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 29
[1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 22
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 15
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [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]
=> ? = 9
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 10
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 17
[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 19
[1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 19
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [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]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 17
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 25
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 97%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 97%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 13 = 14 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 41 = 42 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 27 = 28 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 31 = 32 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 22 = 23 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 31 = 32 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 21 = 22 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 25 = 26 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 18 = 19 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 11 = 12 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 16 = 17 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 12 = 13 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 8 = 9 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 27 = 28 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 18 = 19 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 21 = 22 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 15 = 16 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 9 = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 22 = 23 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 15 = 16 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 18 = 19 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 13 = 14 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 12 = 13 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 13 = 14 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 9 = 10 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 11 = 12 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [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]
=> ? = 26 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 34 - 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 21 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 29 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 22 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 15 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 9 - 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]
=> [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]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 10 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 17 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 17 - 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]
=> [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]
=> ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 11 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 19 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 24 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 25 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 25 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [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]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 24 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 19 - 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]
=> [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]
=> ? = 11 - 1
[]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 26 - 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 34 - 1
[1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 21 - 1
[1,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 29 - 1
[1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 22 - 1
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 15 - 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 10 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 17 - 1
[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 19 - 1
[1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 19 - 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 17 - 1
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0,0]
=> [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]
=> ? = 25 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St001313
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 95%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 95%
Values
[1,0]
=> []
=> => => ? = 1
[1,0,1,0]
=> [1]
=> 10 => 01 => 2
[1,1,0,0]
=> []
=> => => ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 5
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 011 => 3
[1,1,0,0,1,0]
=> [2]
=> 100 => 001 => 3
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 2
[1,1,1,0,0,0]
=> []
=> => => ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 14
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 01011 => 9
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 011001 => 10
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01101 => 7
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 4
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 00101 => 9
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0011 => 6
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 01001 => 7
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 5
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 011 => 3
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0001 => 4
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 001 => 3
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 2
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 01010101 => 42
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0101011 => 28
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 01011001 => 32
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0101101 => 23
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 010111 => 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 01100101 => 32
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0110011 => 22
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 01101001 => 26
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0110101 => 19
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 011011 => 12
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 01110001 => 17
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0111001 => 13
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 011101 => 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 01111 => 5
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0010101 => 28
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 001011 => 19
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0011001 => 22
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 001101 => 16
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 00111 => 10
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0100101 => 23
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 010011 => 16
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0101001 => 19
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 14
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 01011 => 9
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0110001 => 13
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 011001 => 10
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01101 => 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 4
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 000101 => 14
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 00011 => 10
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 001001 => 12
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 00101 => 9
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 1000101110 => 0111010001 => ? = 42
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1000011110 => 0111100001 => ? = 26
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> 1001011110 => 0111101001 => ? = 40
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,1,1,1,1]
=> 10000111110 => 01111100001 => ? = 31
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,1,1,1,1]
=> 1000111110 => 0111110001 => ? = 25
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,1,1]
=> 1000101110 => 0111010001 => ? = 42
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1]
=> 10000011110 => 01111000001 => ? = 31
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,1,1,1]
=> 1000011110 => 0111100001 => ? = 26
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,1]
=> 1000010110 => 0110100001 => ? = 40
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1]
=> 1000001110 => 0111000001 => ? = 25
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => => ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1,1,1,1]
=> 1010111110 => 0111110101 => ? = 34
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,1,1,1,1,1]
=> 10001111110 => 01111110001 => ? = 29
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0111111001 => ? = 22
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,1,1,1]
=> 1001011110 => 0111101001 => ? = 40
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,1,1,1,1]
=> 10000111110 => 01111100001 => ? = 31
[1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,1,1,1,1]
=> 1000111110 => 0111110001 => ? = 25
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,1,1]
=> 1000101110 => 0111010001 => ? = 42
[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,1,1,1]
=> 10000011110 => 01111000001 => ? = 31
[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,1,1,1]
=> 1000011110 => 0111100001 => ? = 26
[1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,1]
=> 1000010110 => 0110100001 => ? = 40
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1]
=> 10000001110 => 01110000001 => ? = 29
[1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,1,1]
=> 1000001110 => 0111000001 => ? = 25
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1]
=> 1000001010 => 0101000001 => ? = 34
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1]
=> 1000000110 => 0110000001 => ? = 22
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> => => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9]
=> 1000000000 => 0000000001 => ? = 10
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1]
=> 1000000010 => 0100000001 => ? = 17
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111101 => ? = 17
[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]
=> 1111111110 => 0111111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10]
=> 10000000000 => 00000000001 => ? = 11
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1]
=> 10000000010 => 01000000001 => ? = 19
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,2]
=> 1000000100 => 0010000001 => ? = 24
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1]
=> 10000000110 => 01100000001 => ? = 25
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,1,1,1,1,1,1]
=> 10011111110 => 01111111001 => ? = 25
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1,1]
=> 1101111110 => 0111111011 => ? = 24
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01111111101 => ? = 19
[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]
=> 11111111110 => 01111111111 => ? = 11
[]
=> []
=> => => ? = 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,1,1,1,1]
=> 1010111110 => 0111110101 => ? = 34
[1,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [4,1,1,1,1,1,1]
=> 10001111110 => 01111110001 => ? = 29
[1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0111111001 => ? = 22
[1,1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,1,1,1,1]
=> 10000111110 => 01111100001 => ? = 31
[1,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [4,1,1,1,1,1]
=> 1000111110 => 0111110001 => ? = 25
Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating 1's as steps (1,0) and 0's as steps (0,1). Given a binary word w, this statistic counts the number of lattice paths from the origin to the same endpoint as w that stay weakly above w.
See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St000070
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 79%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 79%
Values
[1,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[1,1,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[1,1,1,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 14
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 10
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
[1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ? = 42
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> 28
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> 32
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 23
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> 32
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 22
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> 26
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 19
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 12
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 17
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> 28
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> 19
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 22
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 16
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 23
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 16
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 19
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 14
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 10
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 12
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
=> ? = 34
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[3,3,2,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
=> ? = 37
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[4,2,2,1,1],[]]
=> ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? = 42
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[4,3,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? = 41
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(7,9),(8,4),(8,9)],10)
=> ? = 42
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
=> ? = 31
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[4,2,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10)
=> ? = 35
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[3,3,3,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 30
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ? = 42
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [[5,2,2,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? = 41
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[4,4,1,1],[]]
=> ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10)
=> ? = 35
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[5,3,1,1],[]]
=> ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? = 42
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[4,3,3],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 30
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[4,4,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
=> ? = 31
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [[5,3,2],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
=> ? = 37
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [[5,4,1],[]]
=> ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
=> ? = 34
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,1,1]
=> [[2,2,2,2,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,1),(4,3),(4,8),(5,4),(5,9),(6,5),(6,7),(7,9),(9,8)],10)
=> ? = 25
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,2,1,1,1]
=> [[3,2,2,1,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(4,9),(5,2),(6,4),(6,8),(7,1),(7,8),(8,9)],10)
=> ? = 37
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1,1,1]
=> [[3,3,1,1,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,5),(4,3),(5,1),(6,4),(6,8),(7,2),(7,8),(8,9)],10)
=> ? = 34
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> [[4,2,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(7,9),(8,4),(8,9)],10)
=> ? = 40
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,1,1,1,1]
=> [[5,1,1,1,1,1],[]]
=> ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10)
=> ? = 31
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [[2,2,2,2,2],[]]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 21
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
=> ? = 34
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1,1]
=> [[3,3,2,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
=> ? = 37
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,2,1,1]
=> [[4,2,2,1,1],[]]
=> ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? = 42
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,1,1]
=> [[4,3,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? = 41
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(7,9),(8,4),(8,9)],10)
=> ? = 42
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1]
=> [[6,1,1,1,1],[]]
=> ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10)
=> ? = 31
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
=> ? = 31
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2]
=> [[4,2,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10)
=> ? = 35
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1]
=> [[3,3,3,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 30
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ? = 42
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1]
=> [[5,2,2,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? = 41
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [4,4,1,1]
=> [[4,4,1,1],[]]
=> ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10)
=> ? = 35
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,1]
=> [[5,3,1,1],[]]
=> ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? = 42
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,1]
=> [[6,2,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(7,9),(8,4),(8,9)],10)
=> ? = 40
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,3,3]
=> [[4,3,3],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 30
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [[4,4,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
=> ? = 31
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [5,3,2]
=> [[5,3,2],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
=> ? = 37
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,2,2]
=> [[6,2,2],[]]
=> ([(0,6),(0,7),(2,9),(3,5),(4,3),(5,1),(6,4),(6,8),(7,2),(7,8),(8,9)],10)
=> ? = 34
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,4,1]
=> [[5,4,1],[]]
=> ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
=> ? = 34
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,1]
=> [[6,3,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(4,9),(5,2),(6,4),(6,8),(7,1),(7,8),(8,9)],10)
=> ? = 37
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [[5,5],[]]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 21
Description
The number of antichains in a poset.
An antichain in a poset P is a subset of elements of P which are pairwise incomparable.
An order ideal is a subset I of P such that a∈I and b≤Pa implies b∈I. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St001389
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 5
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 14
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 9
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 9
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 6
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 7
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 42
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 28
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 32
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 23
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 32
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> 22
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> 26
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 19
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 12
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 17
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 28
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> 19
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 22
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> 16
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> 23
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> 16
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 19
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 14
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 4
[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]
=> [1,0,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 26
[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]
=> [1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,2,2,2,2,1]
=> ? = 34
[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]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,2,2,2,2,2,1]
=> ? = 29
[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]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,2,2,1,1]
=> ? = 25
[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]
=> ?
=> ?
=> ? = 37
[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]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 22
[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]
=> ?
=> ?
=> ? = 34
[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]
=> ?
=> ?
=> ? = 40
[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]
=> [1,0,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ?
=> ? = 18
[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]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ?
=> ? = 31
[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]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ? = 25
[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]
=> [1,0,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,1,1,1]
=> ? = 21
[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ?
=> ? = 34
[1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ?
=> ? = 37
[1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ?
=> ?
=> ? = 42
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ?
=> ? = 30
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 18
[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ?
=> ? = 41
[1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> ?
=> ? = 28
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ?
=> ? = 42
[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ?
=> ?
=> ? = 33
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ?
=> ? = 15
[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]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ?
=> ? = 31
[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ?
=> ?
=> ? = 21
[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> ?
=> ? = 16
[1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [4,4,3,3,1,1,1,1]
=> ? = 31
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0,0]
=> ?
=> ?
=> ? = 35
[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> ?
=> ? = 25
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? = 15
[1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 30
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,1,1,1,1]
=> ? = 42
[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ?
=> ? = 41
[1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 32
[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ?
=> ? = 23
[1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0,0]
=> ?
=> ?
=> ? = 35
[1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0,0]
=> ?
=> ?
=> ? = 42
[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]
=> ?
=> ?
=> ? = 40
[1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> ?
=> ?
=> ? = 33
[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> ?
=> ? = 26
[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]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ?
=> ? = 29
[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]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ?
=> ? = 25
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ?
=> ? = 21
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ?
=> ? = 13
[1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> ?
=> ? = 30
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ?
=> ? = 31
[1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [6,4,3,1,1,1,1,1]
=> ? = 37
[1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> ?
=> ? = 19
[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]
=> ?
=> ?
=> ? = 34
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ?
=> ? = 28
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 22
Description
The number of partitions of the same length below the given integer partition.
For a partition λ1≥…λk>0, this number is
\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.
Matching statistic: St001232
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 5 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 14 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 9 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 10 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 7 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 9 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 6 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 7 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 5 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 42 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 28 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 32 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 23 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 14 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 32 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 22 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 26 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 19 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 12 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 17 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 13 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 9 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 28 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 19 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 22 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 16 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 23 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 16 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 19 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 14 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 9 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 13 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 10 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 7 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 14 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 10 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 12 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 9 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 9 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 7 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 34 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 20 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 37 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 42 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 30 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 18 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 41 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!