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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St001312
St001312: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => 2
[2] => 1
[1,1,1] => 5
[1,2] => 3
[2,1] => 3
[3] => 1
[1,1,1,1] => 14
[1,1,2] => 9
[1,2,1] => 10
[1,3] => 4
[2,1,1] => 9
[2,2] => 6
[3,1] => 4
[4] => 1
[1,1,1,1,1] => 42
[1,1,1,2] => 28
[1,1,2,1] => 32
[1,1,3] => 14
[1,2,1,1] => 32
[1,2,2] => 22
[1,3,1] => 17
[1,4] => 5
[2,1,1,1] => 28
[2,1,2] => 19
[2,2,1] => 22
[2,3] => 10
[3,1,1] => 14
[3,2] => 10
[4,1] => 5
[5] => 1
[1,1,1,1,1,1] => 132
[1,1,1,1,2] => 90
[1,1,1,2,1] => 104
[1,1,1,3] => 48
[1,1,2,1,1] => 107
[1,1,2,2] => 75
[1,1,3,1] => 62
[1,1,4] => 20
[1,2,1,1,1] => 104
[1,2,1,2] => 72
[1,2,2,1] => 84
[1,2,3] => 40
[1,3,1,1] => 62
[1,3,2] => 45
[1,4,1] => 26
[1,5] => 6
[2,1,1,1,1] => 90
[2,1,1,2] => 62
[2,1,2,1] => 72
[2,1,3] => 34
Description
Number of parabolic noncrossing partitions indexed by the composition.
Also the number of elements in the $\nu$-Tamari lattice with $\nu = \nu_\alpha = 1^{\alpha_1} 0^{\alpha_1} \cdots 1^{\alpha_k} 0^{\alpha_k}$, the bounce path indexed by the composition $\alpha$. These elements are Dyck paths weakly above the bounce path $\nu_\alpha$.
Matching statistic: St000420
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000420: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 5
[1,2] => [1,0,1,1,0,0]
=> 3
[2,1] => [1,1,0,0,1,0]
=> 3
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 14
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 9
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 10
[1,3] => [1,0,1,1,1,0,0,0]
=> 4
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 9
[2,2] => [1,1,0,0,1,1,0,0]
=> 6
[3,1] => [1,1,1,0,0,0,1,0]
=> 4
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 42
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 28
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 32
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 14
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 32
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 22
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 17
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 28
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 19
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 22
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 10
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 14
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 10
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 132
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 90
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 104
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 48
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 107
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 75
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 62
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 20
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 104
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 72
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 84
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 40
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 62
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 45
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 26
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 90
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 62
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 72
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 34
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000419: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> 1 = 2 - 1
[2] => [1,1,0,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 13 = 14 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 41 = 42 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 27 = 28 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 31 = 32 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 13 = 14 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 31 = 32 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 21 = 22 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 16 = 17 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 27 = 28 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 18 = 19 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 21 = 22 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 9 = 10 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 9 = 10 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 131 = 132 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 89 = 90 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 103 = 104 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 47 = 48 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 106 = 107 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 74 = 75 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 61 = 62 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 19 = 20 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 103 = 104 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 71 = 72 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 83 = 84 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 39 = 40 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 61 = 62 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 44 = 45 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 25 = 26 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 89 = 90 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 61 = 62 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 71 = 72 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 33 = 34 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St000108
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 35%●distinct values known / distinct values provided: 31%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 35%●distinct values known / distinct values provided: 31%
Values
[1,1] => [1,0,1,0]
=> [1]
=> 2
[2] => [1,1,0,0]
=> []
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 5
[1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 3
[2,1] => [1,1,0,0,1,0]
=> [2]
=> 3
[3] => [1,1,1,0,0,0]
=> []
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 14
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 9
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 10
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 4
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 9
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 6
[3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 4
[4] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 42
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 28
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 32
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 14
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 32
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 22
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 17
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 28
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 19
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 22
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 14
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 10
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 5
[5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 20
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 26
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 6
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 15
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 20
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 20
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 15
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 6
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> 7
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> 21
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> 21
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 7
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 1
Description
The number of partitions contained in the given partition.
Matching statistic: St001313
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 14%
Mp00093: Dyck paths —to binary word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 14%
Values
[1,1] => [1,0,1,0]
=> 1010 => 2
[2] => [1,1,0,0]
=> 1100 => 1
[1,1,1] => [1,0,1,0,1,0]
=> 101010 => 5
[1,2] => [1,0,1,1,0,0]
=> 101100 => 3
[2,1] => [1,1,0,0,1,0]
=> 110010 => 3
[3] => [1,1,1,0,0,0]
=> 111000 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => 14
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => 9
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => 10
[1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 4
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => 9
[2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => 6
[3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 4
[4] => [1,1,1,1,0,0,0,0]
=> 11110000 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? ∊ {1,5,5,10,10,14,14,17,19,22,22,28,28,32,32,42}
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 101011100010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 101100101010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 101100101100 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 101100110010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 101100111000 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 101110001010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 101110001100 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 110010101100 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 110010110010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 110010111000 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 110011001010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 110011001100 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 110011100010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 111000110010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 111000111000 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 111100001100 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => ? ∊ {1,6,6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? ∊ {1,7,7,21,21,27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {1,7,7,21,21,27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
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: St001232
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 11%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 11%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 5 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {6,9,9,10,14} - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {6,9,9,10,14} - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {6,9,9,10,14} - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {6,9,9,10,14} - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {6,9,9,10,14} - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[1,1,3] => [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]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[3,2] => [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]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[2,1,3] => [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]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[1,6] => [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]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
Values
[1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,5} + 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,5} + 1
[3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,9,9,10,14} + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,9,9,10,14} + 1
[1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,9,9,10,14} + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,9,9,10,14} + 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,9,9,10,14} + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,9,9,10,14} + 1
[4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,10,10,14,14,17,19,22,22,28,28,32,32,42} + 1
[5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[1,5] => ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {6,15,15,20,20,20,26,34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132} + 1
[6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8 = 7 + 1
[7] => ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
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