searching the database
Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001312
St001312: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => 2 = 1 + 1
[2] => 1 = 0 + 1
[1,1,1] => 5 = 4 + 1
[1,2] => 3 = 2 + 1
[2,1] => 3 = 2 + 1
[3] => 1 = 0 + 1
[1,1,1,1] => 14 = 13 + 1
[1,1,2] => 9 = 8 + 1
[1,2,1] => 10 = 9 + 1
[1,3] => 4 = 3 + 1
[2,1,1] => 9 = 8 + 1
[2,2] => 6 = 5 + 1
[3,1] => 4 = 3 + 1
[4] => 1 = 0 + 1
[1,1,1,1,1] => 42 = 41 + 1
[1,1,1,2] => 28 = 27 + 1
[1,1,2,1] => 32 = 31 + 1
[1,1,3] => 14 = 13 + 1
[1,2,1,1] => 32 = 31 + 1
[1,2,2] => 22 = 21 + 1
[1,3,1] => 17 = 16 + 1
[1,4] => 5 = 4 + 1
[2,1,1,1] => 28 = 27 + 1
[2,1,2] => 19 = 18 + 1
[2,2,1] => 22 = 21 + 1
[2,3] => 10 = 9 + 1
[3,1,1] => 14 = 13 + 1
[3,2] => 10 = 9 + 1
[4,1] => 5 = 4 + 1
[5] => 1 = 0 + 1
[1,1,1,1,1,1] => 132 = 131 + 1
[1,1,1,1,2] => 90 = 89 + 1
[1,1,1,2,1] => 104 = 103 + 1
[1,1,1,3] => 48 = 47 + 1
[1,1,2,1,1] => 107 = 106 + 1
[1,1,2,2] => 75 = 74 + 1
[1,1,3,1] => 62 = 61 + 1
[1,1,4] => 20 = 19 + 1
[1,2,1,1,1] => 104 = 103 + 1
[1,2,1,2] => 72 = 71 + 1
[1,2,2,1] => 84 = 83 + 1
[1,2,3] => 40 = 39 + 1
[1,3,1,1] => 62 = 61 + 1
[1,3,2] => 45 = 44 + 1
[1,4,1] => 26 = 25 + 1
[1,5] => 6 = 5 + 1
[2,1,1,1,1] => 90 = 89 + 1
[2,1,1,2] => 62 = 61 + 1
[2,1,2,1] => 72 = 71 + 1
[2,1,3] => 34 = 33 + 1
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: St000419
(load all 4 compositions to match this statistic)
(load all 4 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] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 4
[1,2] => [1,0,1,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 13
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 8
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 9
[1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 8
[2,2] => [1,1,0,0,1,1,0,0]
=> 5
[3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 41
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 27
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 31
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 13
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 31
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 21
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 16
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 27
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 18
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 21
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 9
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 13
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 9
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 131
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 89
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 103
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 47
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 106
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 74
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 61
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 19
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 103
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 71
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 83
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 39
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 61
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 44
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 25
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 89
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 61
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 71
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 33
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St000420
(load all 3 compositions to match this statistic)
(load all 3 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 = 1 + 1
[2] => [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 14 = 13 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 9 = 8 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 10 = 9 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 9 = 8 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 42 = 41 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 28 = 27 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 32 = 31 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 14 = 13 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 32 = 31 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 22 = 21 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 17 = 16 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 28 = 27 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 19 = 18 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 22 = 21 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 10 = 9 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 14 = 13 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 10 = 9 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 132 = 131 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 90 = 89 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 104 = 103 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 48 = 47 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 107 = 106 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 75 = 74 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 62 = 61 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 20 = 19 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 104 = 103 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 72 = 71 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 84 = 83 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 40 = 39 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 62 = 61 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 45 = 44 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 26 = 25 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 90 = 89 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 62 = 61 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 72 = 71 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 34 = 33 + 1
Description
The number of Dyck paths that are weakly above a Dyck path.
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 = 1 + 1
[2] => [1,1,0,0]
=> []
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 5 = 4 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 3 = 2 + 1
[2,1] => [1,1,0,0,1,0]
=> [2]
=> 3 = 2 + 1
[3] => [1,1,1,0,0,0]
=> []
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 14 = 13 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 9 = 8 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 10 = 9 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 4 = 3 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 9 = 8 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 6 = 5 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 4 = 3 + 1
[4] => [1,1,1,1,0,0,0,0]
=> []
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 42 = 41 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 28 = 27 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 32 = 31 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 14 = 13 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 32 = 31 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 22 = 21 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 17 = 16 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 5 = 4 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 28 = 27 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 19 = 18 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 22 = 21 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10 = 9 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 14 = 13 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 10 = 9 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 5 = 4 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1 = 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]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 20 = 19 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 26 = 25 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 6 = 5 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 15 = 14 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? ∊ {33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 20 = 19 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 20 = 19 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 15 = 14 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 6 = 5 + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 1 = 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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[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]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> ? ∊ {26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> 7 = 6 + 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> 21 = 20 + 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> 21 = 20 + 1
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 7 = 6 + 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 1 = 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 = 1 + 1
[2] => [1,1,0,0]
=> 1100 => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 101010 => 5 = 4 + 1
[1,2] => [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[2,1] => [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[3] => [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => 14 = 13 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => 9 = 8 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => 10 = 9 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => 9 = 8 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => 6 = 5 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 4 = 3 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 11110000 => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? ∊ {0,4,4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 101011100010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 101100101010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 101100101100 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 101100110010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 101100111000 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 101110001010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 101110001100 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 110010101100 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 110010110010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 110010111000 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 110011001010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 110011001100 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 110011100010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 111000110010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 111000111000 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 111100001100 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => ? ∊ {0,5,5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? ∊ {0,6,6,20,20,26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {0,6,6,20,20,26,26,34,34,36,54,54,64,64,68,74,74,80,80,94,94,104,104,106,106,116,116,125,136,136,144,144,156,156,164,164,170,170,179,179,184,184,206,210,210,218,218,232,240,240,251,251,254,254,280,294,294,296,296,344,344,358,358,428} + 1
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] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 4
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,8,8,9,13}
[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]
=> ? ∊ {5,8,8,9,13}
[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]
=> ? ∊ {5,8,8,9,13}
[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
[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]
=> ? ∊ {5,8,8,9,13}
[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]
=> ? ∊ {5,8,8,9,13}
[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] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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]
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41}
[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] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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]
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131}
[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
[1,6] => [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] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [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
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 = 1 + 2
[2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,4} + 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,4} + 2
[3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[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 = 3 + 2
[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)
=> ? ∊ {3,5,8,8,9,13} + 2
[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)
=> ? ∊ {3,5,8,8,9,13} + 2
[1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5,8,8,9,13} + 2
[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)
=> ? ∊ {3,5,8,8,9,13} + 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5,8,8,9,13} + 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5,8,8,9,13} + 2
[4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[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 = 4 + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[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)
=> ? ∊ {4,9,9,13,13,16,18,21,21,27,27,31,31,41} + 2
[5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[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 = 5 + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[1,5] => ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[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)
=> ? ∊ {5,14,14,19,19,19,25,33,33,39,39,44,44,47,47,52,61,61,61,71,71,74,74,83,89,89,103,103,106,131} + 2
[6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[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 = 6 + 2
[7] => ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
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.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!