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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001389
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 3
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 5
[3,1,1]
=> 3
[2,2,1]
=> 3
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 7
[4,1,1]
=> 4
[3,3]
=> 6
[3,2,1]
=> 5
[3,1,1,1]
=> 3
[2,2,2]
=> 4
[2,2,1,1]
=> 3
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 7
[6,1]
=> 6
[5,2]
=> 9
[5,1,1]
=> 5
[4,3]
=> 9
[4,2,1]
=> 7
[4,1,1,1]
=> 4
[3,3,1]
=> 6
[3,2,2]
=> 7
[3,2,1,1]
=> 5
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 4
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 7
[6,2]
=> 11
[6,1,1]
=> 6
[5,3]
=> 12
[5,2,1]
=> 9
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: St000420
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 92%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 92%
Values
[1]
=> [1]
=> []
=> []
=> ? = 1
[2]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1]
=> [2]
=> []
=> []
=> ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,1]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 9
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 9
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 7
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 8
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 11
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 12
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 9
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 10
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 9
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 9
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 1
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 11
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 17
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 12
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 11
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 19
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [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]
=> ? = 24
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 17
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 1
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13
[12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 12
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 21
[11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 11
[10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 27
[10,2,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 19
[10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [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]
=> ? = 30
[9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> [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]
=> ? = 24
[9,2,2]
=> [3,3,1,1,1,1,1,1,1]
=> [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]
=> ? = 25
[9,2,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 17
[9,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> []
=> []
=> ? = 1
[14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 14
[13,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 23
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 12
[11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 30
[11,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 21
[11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 11
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 34
[10,3,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 27
[10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 28
[10,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 19
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 92%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 92%
Values
[1]
=> [1]
=> []
=> []
=> ? = 1 - 1
[2]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> []
=> ? = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1 - 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 8 = 9 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7 = 8 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 10 = 11 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 11 = 12 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 8 = 9 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 9 = 10 - 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1 - 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 1 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10 - 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 1 - 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 11 - 1
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10 - 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 17 - 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 1 - 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 12 - 1
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 11 - 1
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 19 - 1
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10 - 1
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [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]
=> ? = 24 - 1
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 17 - 1
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 1 - 1
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13 - 1
[12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 12 - 1
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 21 - 1
[11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 11 - 1
[10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 27 - 1
[10,2,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 19 - 1
[10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 10 - 1
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [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]
=> ? = 30 - 1
[9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> [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]
=> ? = 24 - 1
[9,2,2]
=> [3,3,1,1,1,1,1,1,1]
=> [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]
=> ? = 25 - 1
[9,2,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 17 - 1
[9,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> []
=> []
=> ? = 1 - 1
[14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 14 - 1
[13,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13 - 1
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 23 - 1
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 12 - 1
[11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 30 - 1
[11,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 21 - 1
[11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 11 - 1
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 34 - 1
[10,3,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 27 - 1
[10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 28 - 1
[10,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 19 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 71%●distinct values known / distinct values provided: 48%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 71%●distinct values known / distinct values provided: 48%
Values
[1]
=> [1]
=> []
=> 1
[2]
=> [1,1]
=> [1]
=> 2
[1,1]
=> [2]
=> []
=> 1
[3]
=> [1,1,1]
=> [1,1]
=> 3
[2,1]
=> [2,1]
=> [1]
=> 2
[1,1,1]
=> [3]
=> []
=> 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 4
[3,1]
=> [2,1,1]
=> [1,1]
=> 3
[2,2]
=> [2,2]
=> [2]
=> 3
[2,1,1]
=> [3,1]
=> [1]
=> 2
[1,1,1,1]
=> [4]
=> []
=> 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 4
[3,2]
=> [2,2,1]
=> [2,1]
=> 5
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 3
[2,2,1]
=> [3,2]
=> [2]
=> 3
[2,1,1,1]
=> [4,1]
=> [1]
=> 2
[1,1,1,1,1]
=> [5]
=> []
=> 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 5
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 7
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 4
[3,3]
=> [2,2,2]
=> [2,2]
=> 6
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 5
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 3
[2,2,2]
=> [3,3]
=> [3]
=> 4
[2,2,1,1]
=> [4,2]
=> [2]
=> 3
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 2
[1,1,1,1,1,1]
=> [6]
=> []
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 7
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 9
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 5
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 9
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 7
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 4
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 6
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 7
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 5
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 3
[2,2,2,1]
=> [4,3]
=> [3]
=> 4
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 3
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> []
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 8
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 7
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 11
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 12
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 9
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
[12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 21
[10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 27
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> ? = 30
[8,5]
=> [2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> ? = 30
[7,6]
=> [2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> ? = 27
[14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
[13,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 23
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
[11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> ? = 30
[11,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 21
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> ? = 34
[10,3,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 27
[10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? = 28
[9,5]
=> [2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> ? = 35
[9,4,1]
=> [3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> ? = 30
[9,3,2]
=> [3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> ? = 39
[8,6]
=> [2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> ? = 33
[8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> ? = 30
[8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> ? = 44
[8,3,3]
=> [3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> ? = 40
[7,7]
=> [2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> ? = 28
[7,6,1]
=> [3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> ? = 27
[7,5,2]
=> [3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> ? = 43
[7,4,3]
=> [3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> ? = 46
[6,6,2]
=> [3,3,2,2,2,2]
=> [3,2,2,2,2]
=> ? = 36
[6,5,3]
=> [3,3,3,2,2,1]
=> [3,3,2,2,1]
=> ? = 43
[6,4,4]
=> [3,3,3,3,1,1]
=> [3,3,3,1,1]
=> ? = 40
[5,5,4]
=> [3,3,3,3,2]
=> [3,3,3,2]
=> ? = 34
[15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 15
[14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
[13,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 25
[13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
[12,3]
=> [2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> ? = 33
[12,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 23
[12,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
[11,4]
=> [2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> ? = 38
[11,3,1]
=> [3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> ? = 30
[11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> ? = 31
[11,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 21
[10,5]
=> [2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> ? = 40
[10,4,1]
=> [3,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> ? = 34
[10,3,2]
=> [3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> ? = 44
[10,3,1,1]
=> [4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 27
[10,2,2,1]
=> [4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? = 28
[9,6]
=> [2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 39
[9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> ? = 35
Description
The number of partitions contained in the given partition.
Matching statistic: St001313
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 58%
Mp00104: Binary words —reverse⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 58%
Values
[1]
=> 10 => 01 => 10 => 1
[2]
=> 100 => 001 => 010 => 2
[1,1]
=> 110 => 011 => 110 => 1
[3]
=> 1000 => 0001 => 0010 => 3
[2,1]
=> 1010 => 0101 => 1010 => 2
[1,1,1]
=> 1110 => 0111 => 1110 => 1
[4]
=> 10000 => 00001 => 00010 => 4
[3,1]
=> 10010 => 01001 => 10010 => 3
[2,2]
=> 1100 => 0011 => 0110 => 3
[2,1,1]
=> 10110 => 01101 => 11010 => 2
[1,1,1,1]
=> 11110 => 01111 => 11110 => 1
[5]
=> 100000 => 000001 => 000010 => 5
[4,1]
=> 100010 => 010001 => 100010 => 4
[3,2]
=> 10100 => 00101 => 01010 => 5
[3,1,1]
=> 100110 => 011001 => 110010 => 3
[2,2,1]
=> 11010 => 01011 => 10110 => 3
[2,1,1,1]
=> 101110 => 011101 => 111010 => 2
[1,1,1,1,1]
=> 111110 => 011111 => 111110 => 1
[6]
=> 1000000 => 0000001 => 0000010 => 6
[5,1]
=> 1000010 => 0100001 => 1000010 => 5
[4,2]
=> 100100 => 001001 => 010010 => 7
[4,1,1]
=> 1000110 => 0110001 => 1100010 => 4
[3,3]
=> 11000 => 00011 => 00110 => 6
[3,2,1]
=> 101010 => 010101 => 101010 => 5
[3,1,1,1]
=> 1001110 => 0111001 => 1110010 => 3
[2,2,2]
=> 11100 => 00111 => 01110 => 4
[2,2,1,1]
=> 110110 => 011011 => 110110 => 3
[2,1,1,1,1]
=> 1011110 => 0111101 => 1111010 => 2
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 1111110 => 1
[7]
=> 10000000 => 00000001 => 00000010 => 7
[6,1]
=> 10000010 => 01000001 => 10000010 => 6
[5,2]
=> 1000100 => 0010001 => 0100010 => 9
[5,1,1]
=> 10000110 => 01100001 => 11000010 => 5
[4,3]
=> 101000 => 000101 => 001010 => 9
[4,2,1]
=> 1001010 => 0101001 => 1010010 => 7
[4,1,1,1]
=> 10001110 => 01110001 => 11100010 => 4
[3,3,1]
=> 110010 => 010011 => 100110 => 6
[3,2,2]
=> 101100 => 001101 => 011010 => 7
[3,2,1,1]
=> 1010110 => 0110101 => 1101010 => 5
[3,1,1,1,1]
=> 10011110 => 01111001 => 11110010 => 3
[2,2,2,1]
=> 111010 => 010111 => 101110 => 4
[2,2,1,1,1]
=> 1101110 => 0111011 => 1110110 => 3
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 11111010 => 2
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 11111110 => 1
[8]
=> 100000000 => 000000001 => 000000010 => 8
[7,1]
=> 100000010 => 010000001 => 100000010 => 7
[6,2]
=> 10000100 => 00100001 => 01000010 => 11
[6,1,1]
=> 100000110 => 011000001 => 110000010 => 6
[5,3]
=> 1001000 => 0001001 => 0010010 => 12
[5,2,1]
=> 10001010 => 01010001 => 10100010 => 9
[9]
=> 1000000000 => 0000000001 => 0000000010 => ? = 9
[8,1]
=> 1000000010 => 0100000001 => 1000000010 => ? = 8
[7,1,1]
=> 1000000110 => 0110000001 => 1100000010 => ? = 7
[6,1,1,1]
=> 1000001110 => 0111000001 => 1110000010 => ? = 6
[5,1,1,1,1]
=> 1000011110 => 0111100001 => 1111000010 => ? = 5
[4,1,1,1,1,1]
=> 1000111110 => 0111110001 => 1111100010 => ? = 4
[3,1,1,1,1,1,1]
=> 1001111110 => 0111111001 => 1111110010 => ? = 3
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111101 => 1111111010 => ? = 2
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => 0111111111 => 1111111110 => ? = 1
[10]
=> 10000000000 => 00000000001 => 00000000010 => ? = 10
[9,1]
=> 10000000010 => 01000000001 => 10000000010 => ? = 9
[8,2]
=> 1000000100 => 0010000001 => 0100000010 => ? = 15
[8,1,1]
=> 10000000110 => 01100000001 => 11000000010 => ? = 8
[7,2,1]
=> 1000001010 => 0101000001 => 1010000010 => ? = 13
[7,1,1,1]
=> 10000001110 => 01110000001 => 11100000010 => ? = 7
[6,2,1,1]
=> 1000010110 => 0110100001 => 1101000010 => ? = 11
[6,1,1,1,1]
=> 10000011110 => 01111000001 => 11110000010 => ? = 6
[5,2,1,1,1]
=> 1000101110 => 0111010001 => 1110100010 => ? = 9
[5,1,1,1,1,1]
=> 10000111110 => 01111100001 => 11111000010 => ? = 5
[4,2,1,1,1,1]
=> 1001011110 => 0111101001 => 1111010010 => ? = 7
[4,1,1,1,1,1,1]
=> 10001111110 => 01111110001 => 11111100010 => ? = 4
[3,2,1,1,1,1,1]
=> 1010111110 => 0111110101 => 1111101010 => ? = 5
[3,1,1,1,1,1,1,1]
=> 10011111110 => 01111111001 => 11111110010 => ? = 3
[2,2,1,1,1,1,1,1]
=> 1101111110 => 0111111011 => 1111110110 => ? = 3
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01111111101 => 11111111010 => ? = 2
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 01111111111 => 11111111110 => ? = 1
[11]
=> 100000000000 => 000000000001 => 000000000010 => ? = 11
[10,1]
=> 100000000010 => 010000000001 => 100000000010 => ? = 10
[9,2]
=> 10000000100 => 00100000001 => 01000000010 => ? = 17
[9,1,1]
=> 100000000110 => 011000000001 => 110000000010 => ? = 9
[8,3]
=> 1000001000 => 0001000001 => 0010000010 => ? = 21
[8,2,1]
=> 10000001010 => 01010000001 => 10100000010 => ? = 15
[8,1,1,1]
=> 100000001110 => 011100000001 => ? => ? = 8
[7,3,1]
=> 1000010010 => 0100100001 => 1001000010 => ? = 18
[7,2,2]
=> 1000001100 => 0011000001 => 0110000010 => ? = 19
[7,2,1,1]
=> 10000010110 => 01101000001 => 11010000010 => ? = 13
[7,1,1,1,1]
=> 100000011110 => 011110000001 => 111100000010 => ? = 7
[6,3,1,1]
=> 1000100110 => 0110010001 => 1100100010 => ? = 15
[6,2,2,1]
=> 1000011010 => 0101100001 => 1011000010 => ? = 16
[6,2,1,1,1]
=> 10000101110 => 01110100001 => 11101000010 => ? = 11
[6,1,1,1,1,1]
=> 100000111110 => 011111000001 => 111110000010 => ? = 6
[5,3,1,1,1]
=> 1001001110 => 0111001001 => 1110010010 => ? = 12
[5,2,2,1,1]
=> 1000110110 => 0110110001 => 1101100010 => ? = 13
[5,2,1,1,1,1]
=> 10001011110 => 01111010001 => 11110100010 => ? = 9
[5,1,1,1,1,1,1]
=> 100001111110 => 011111100001 => 111111000010 => ? = 5
[4,3,1,1,1,1]
=> 1010011110 => 0111100101 => 1111001010 => ? = 9
[4,2,2,1,1,1]
=> 1001101110 => 0111011001 => 1110110010 => ? = 10
[4,2,1,1,1,1,1]
=> 10010111110 => 01111101001 => 11111010010 => ? = 7
[4,1,1,1,1,1,1,1]
=> 100011111110 => 011111110001 => ? => ? = 4
[3,3,1,1,1,1,1]
=> 1100111110 => 0111110011 => ? => ? = 6
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: St000110
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 7
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 6
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1
[7]
=> [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]
=> [7,1,2,3,4,5,6] => 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => 9
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 9
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => 7
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 6
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 7
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => 5
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7] => 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => 7
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,7,5] => 11
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8] => 6
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => 12
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,2,3,6,4,7] => 9
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,2,5,3,6,7] => ? = 7
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => ? = 4
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => ? = 5
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => ? = 3
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => ? = 9
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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] => ? = 8
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => ? = 13
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9] => ? = 7
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,1,2,3,4,7,5,8] => ? = 11
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => ? = 6
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,1,2,3,6,4,7,8] => ? = 9
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8,9] => ? = 5
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3,6,7] => ? = 9
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,1,2,5,6,3,7] => ? = 10
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,2,5,3,6,7,8] => ? = 7
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8,9] => ? = 4
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 7
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7,8] => ? = 5
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9] => ? = 3
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ? = 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 2
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ? = 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => ? = 10
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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] => ? = 9
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => ? = 15
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7,9,10] => ? = 8
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,8,5,6] => ? = 18
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [7,1,2,3,4,5,8,6,9] => ? = 13
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9,10] => ? = 7
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5,8] => ? = 15
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [6,1,2,3,4,7,8,5] => ? = 16
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [6,1,2,3,4,7,5,8,9] => ? = 11
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8,9,10] => ? = 6
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,1,2,6,3,7,4] => ? = 19
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [5,1,2,6,3,4,7,8] => ? = 12
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [5,1,2,3,6,7,4,8] => ? = 13
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [5,1,2,3,6,4,7,8,9] => ? = 9
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8,9,10] => ? = 5
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,1,2,3,6,7] => ? = 10
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3,7] => ? = 14
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,3,6,7,8] => ? = 9
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 13
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,2,5,6,3,7,8] => ? = 10
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,5,3,6,7,8,9] => ? = 7
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8,9,10] => ? = 4
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,1,5,2,6,7] => ? = 9
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,4,5,6,2,7] => ? = 9
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,2,6,7,8] => ? = 7
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7,8,9] => ? = 5
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9,10] => ? = 3
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000883
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 14%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 14%
Values
[1]
=> [[1]]
=> [[1]]
=> [1] => 1
[2]
=> [[1,2]]
=> [[1],[2]]
=> [2,1] => 2
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 1
[3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 3
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => 7
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => 6
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => 5
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 4
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 3
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => 6
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[1,6],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6] => ? = 9
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 5
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => ? = 9
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ? = 7
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 4
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ? = 6
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? = 7
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => ? = 5
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => 3
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => ? = 4
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => 7
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => ? = 11
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => 6
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[1,6],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6] => ? = 12
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,6,8],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8] => ? = 9
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => 5
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5] => ? = 10
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,5,8],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8] => ? = 9
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,5,7],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7] => ? = 10
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,5,7,8],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8] => ? = 7
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => 4
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,4,7],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7] => ? = 9
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => ? = 6
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,4,6,8],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8] => ? = 7
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[1,4,6,7,8],[2,5],[3]]
=> [3,2,5,1,4,6,7,8] => ? = 5
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => 3
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => 5
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => ? = 4
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => ? = 3
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => 2
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => 9
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => 8
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [[1,8],[2,9],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,9,1,8] => ? = 13
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => 7
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [[1,7],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7] => ? = 15
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [[1,7,9],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9] => ? = 11
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => 6
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [[1,6],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6] => ? = 14
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,6,9],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6,9] => ? = 12
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,6,8],[2,7,9],[3],[4],[5]]
=> [5,4,3,2,7,9,1,6,8] => ? = 13
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [[1,6,8,9],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8,9] => ? = 9
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5,9] => ? = 10
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> [4,3,7,2,6,9,1,5,8] => ? = 14
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8,9] => ? = 9
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7,9] => ? = 10
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8,9] => ? = 7
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> [3,6,9,2,5,8,1,4,7] => ? = 10
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7,9] => ? = 9
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8,9] => ? = 6
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> [3,2,5,7,9,1,4,6,8] => ? = 9
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8,9] => ? = 7
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> [3,2,5,1,4,6,7,8,9] => ? = 5
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7,9] => ? = 5
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> [2,4,6,1,3,5,7,8,9] => ? = 4
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,3,5,6,7,8,9],[2,4]]
=> [2,4,1,3,5,6,7,8,9] => ? = 3
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [[1,9],[2,10],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,10,1,9] => ? = 15
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [[1,8],[2,9],[3,10],[4],[5],[6],[7]]
=> [7,6,5,4,3,10,2,9,1,8] => ? = 18
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [[1,8,10],[2,9],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,9,1,8,10] => ? = 13
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [[1,7],[2,8],[3,9],[4,10],[5],[6]]
=> [6,5,4,10,3,9,2,8,1,7] => ? = 18
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [[1,7,10],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7,10] => ? = 15
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [[1,7,9],[2,8,10],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,10,1,7,9] => ? = 16
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [[1,7,9,10],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9,10] => ? = 11
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [5,10,4,9,3,8,2,7,1,6] => ? = 15
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [[1,6,10],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6,10] => ? = 14
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001464
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 19%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 19%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 7
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 6
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1
[7]
=> [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]
=> [7,1,2,3,4,5,6] => ? = 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => 9
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 9
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => 7
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => ? = 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 6
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 7
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => 5
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7] => ? = 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => ? = 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => ? = 7
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 11
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8] => ? = 6
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => 12
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,2,3,6,4,7] => ? = 9
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8] => ? = 5
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 10
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => 9
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => 10
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,2,5,3,6,7] => ? = 7
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => ? = 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 9
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,1,2,5,6] => 6
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 7
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => ? = 5
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => ? = 3
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => 4
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => ? = 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => ? = 2
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => ? = 9
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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] => ? = 8
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => ? = 13
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9] => ? = 7
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,7,4,5] => ? = 15
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,1,2,3,4,7,5,8] => ? = 11
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => ? = 6
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => 14
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,6,3,4,7] => ? = 12
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,1,2,3,6,7,4] => ? = 13
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,1,2,3,6,4,7,8] => ? = 9
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8,9] => ? = 5
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => 10
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => 14
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3,6,7] => ? = 9
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,1,2,5,6,3,7] => ? = 10
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,2,5,3,6,7,8] => ? = 7
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8,9] => ? = 4
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,1,2,5,6,7] => ? = 6
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 7
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7,8] => ? = 5
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9] => ? = 3
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => ? = 4
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ? = 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 2
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ? = 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => ? = 10
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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] => ? = 9
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => ? = 15
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7,9,10] => ? = 8
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,8,5,6] => ? = 18
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [7,1,2,3,4,5,8,6,9] => ? = 13
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9,10] => ? = 7
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,1,2,7,3,4,5] => ? = 18
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St001684
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 11%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 11%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3 = 4 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 4 = 5 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 3 = 4 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 4 = 5 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => ? = 6 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => ? = 5 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 6 = 7 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => ? = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 5 = 6 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => ? = 3 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2 = 3 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => ? = 2 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ? = 1 - 1
[7]
=> [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]
=> [7,1,2,3,4,5,6] => ? = 7 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 6 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => ? = 9 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 5 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 8 = 9 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => ? = 7 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => ? = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 5 = 6 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 6 = 7 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => ? = 5 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7] => ? = 3 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 3 = 4 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => ? = 3 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 2 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => ? = 8 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => ? = 7 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 11 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8] => ? = 6 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => ? = 12 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,2,3,6,4,7] => ? = 9 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8] => ? = 5 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 9 = 10 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => ? = 9 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => ? = 10 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,2,5,3,6,7] => ? = 7 - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => ? = 4 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 8 = 9 - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,1,2,5,6] => ? = 6 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => ? = 7 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => ? = 5 - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => ? = 3 - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4 = 5 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => ? = 4 - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => ? = 3 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => ? = 2 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 1 - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => ? = 9 - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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] => ? = 8 - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => ? = 13 - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9] => ? = 7 - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,7,4,5] => ? = 15 - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,1,2,3,4,7,5,8] => ? = 11 - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => ? = 6 - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => ? = 14 - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,6,3,4,7] => ? = 12 - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,1,2,3,6,7,4] => ? = 13 - 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,1,2,3,6,4,7,8] => ? = 9 - 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8,9] => ? = 5 - 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => ? = 10 - 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => ? = 14 - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 9 = 10 - 1
Description
The reduced word complexity of a permutation.
For a permutation \pi, this is the smallest length of a word in simple transpositions that contains all reduced expressions of \pi.
For example, the permutation [3,2,1] = (12)(23)(12) = (23)(12)(23) and the reduced word complexity is 4 since the smallest words containing those two reduced words as subwords are (12),(23),(12),(23) and also (23),(12),(23),(12).
This statistic appears in [1, Question 6.1].
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 9%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 9%
Values
[1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 5 - 1
[3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 6 - 1
[3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 5 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> [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
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 9 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 9 - 1
[4,2,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 6 - 1
[3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 7 - 1
[3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> [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
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 12 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 9 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 10 - 1
[4,3,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 9 - 1
[4,2,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 10 - 1
[4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 9 - 1
[3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
[3,2,2,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 7 - 1
[3,2,1,1,1]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1,1,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,1,1,1,1,1]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9 - 1
[8,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 - 1
[7,2]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 13 - 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[6,3]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 15 - 1
[6,2,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
[2,2,2,2,1]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2,1,1,1]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,2,2,2]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,2,2,2,1,1]
=> [6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2,2,2,1]
=> [6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,2,2,2,2,2]
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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