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Your data matches 209 different statistics following compositions of up to 3 maps.
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Matching statistic: St000228
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,1] => [1,1]
=> 2
[2] => [2]
=> 2
[1,1,1] => [1,1,1]
=> 3
[1,2] => [2,1]
=> 3
[2,1] => [2,1]
=> 3
[3] => [3]
=> 3
[1,1,1,1] => [1,1,1,1]
=> 4
[1,1,2] => [2,1,1]
=> 4
[1,2,1] => [2,1,1]
=> 4
[1,3] => [3,1]
=> 4
[2,1,1] => [2,1,1]
=> 4
[2,2] => [2,2]
=> 4
[3,1] => [3,1]
=> 4
[4] => [4]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> 5
[1,1,1,2] => [2,1,1,1]
=> 5
[1,1,2,1] => [2,1,1,1]
=> 5
[1,1,3] => [3,1,1]
=> 5
[1,2,1,1] => [2,1,1,1]
=> 5
[1,2,2] => [2,2,1]
=> 5
[1,3,1] => [3,1,1]
=> 5
[1,4] => [4,1]
=> 5
[2,1,1,1] => [2,1,1,1]
=> 5
[2,1,2] => [2,2,1]
=> 5
[2,2,1] => [2,2,1]
=> 5
[2,3] => [3,2]
=> 5
[3,1,1] => [3,1,1]
=> 5
[3,2] => [3,2]
=> 5
[4,1] => [4,1]
=> 5
[5] => [5]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [2,1,1,1,1]
=> 6
[1,1,1,2,1] => [2,1,1,1,1]
=> 6
[1,1,1,3] => [3,1,1,1]
=> 6
[1,1,2,1,1] => [2,1,1,1,1]
=> 6
[1,1,2,2] => [2,2,1,1]
=> 6
[1,1,3,1] => [3,1,1,1]
=> 6
[1,1,4] => [4,1,1]
=> 6
[1,2,1,1,1] => [2,1,1,1,1]
=> 6
[1,2,1,2] => [2,2,1,1]
=> 6
[1,2,2,1] => [2,2,1,1]
=> 6
[1,2,3] => [3,2,1]
=> 6
[1,3,1,1] => [3,1,1,1]
=> 6
[1,3,2] => [3,2,1]
=> 6
[1,4,1] => [4,1,1]
=> 6
[1,5] => [5,1]
=> 6
[2,1,1,1,1] => [2,1,1,1,1]
=> 6
[2,1,1,2] => [2,2,1,1]
=> 6
[2,1,2,1] => [2,2,1,1]
=> 6
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000395
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> 3
[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]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> 4
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[2,2] => [1,1,0,0,1,1,0,0]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> 4
[4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[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]
=> 5
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 5
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 5
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 5
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 6
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 6
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 6
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 6
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[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]
=> 6
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001020
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> 3
[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]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> 4
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[2,2] => [1,1,0,0,1,1,0,0]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> 4
[4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[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]
=> 5
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 5
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 5
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 5
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 6
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 6
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 6
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 6
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[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]
=> 6
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 6
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000998
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 2 = 1 + 1
[1,1] => [1,0,1,0]
=> 3 = 2 + 1
[2] => [1,1,0,0]
=> 3 = 2 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2] => [1,0,1,1,0,0]
=> 4 = 3 + 1
[2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 6 = 5 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6 = 5 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6 = 5 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 7 = 6 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000293
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 1
[1,1] => [1,1]
=> 110 => 2
[2] => [2]
=> 100 => 2
[1,1,1] => [1,1,1]
=> 1110 => 3
[1,2] => [2,1]
=> 1010 => 3
[2,1] => [2,1]
=> 1010 => 3
[3] => [3]
=> 1000 => 3
[1,1,1,1] => [1,1,1,1]
=> 11110 => 4
[1,1,2] => [2,1,1]
=> 10110 => 4
[1,2,1] => [2,1,1]
=> 10110 => 4
[1,3] => [3,1]
=> 10010 => 4
[2,1,1] => [2,1,1]
=> 10110 => 4
[2,2] => [2,2]
=> 1100 => 4
[3,1] => [3,1]
=> 10010 => 4
[4] => [4]
=> 10000 => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 5
[1,1,1,2] => [2,1,1,1]
=> 101110 => 5
[1,1,2,1] => [2,1,1,1]
=> 101110 => 5
[1,1,3] => [3,1,1]
=> 100110 => 5
[1,2,1,1] => [2,1,1,1]
=> 101110 => 5
[1,2,2] => [2,2,1]
=> 11010 => 5
[1,3,1] => [3,1,1]
=> 100110 => 5
[1,4] => [4,1]
=> 100010 => 5
[2,1,1,1] => [2,1,1,1]
=> 101110 => 5
[2,1,2] => [2,2,1]
=> 11010 => 5
[2,2,1] => [2,2,1]
=> 11010 => 5
[2,3] => [3,2]
=> 10100 => 5
[3,1,1] => [3,1,1]
=> 100110 => 5
[3,2] => [3,2]
=> 10100 => 5
[4,1] => [4,1]
=> 100010 => 5
[5] => [5]
=> 100000 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => 6
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => 6
[1,1,1,2,1] => [2,1,1,1,1]
=> 1011110 => 6
[1,1,1,3] => [3,1,1,1]
=> 1001110 => 6
[1,1,2,1,1] => [2,1,1,1,1]
=> 1011110 => 6
[1,1,2,2] => [2,2,1,1]
=> 110110 => 6
[1,1,3,1] => [3,1,1,1]
=> 1001110 => 6
[1,1,4] => [4,1,1]
=> 1000110 => 6
[1,2,1,1,1] => [2,1,1,1,1]
=> 1011110 => 6
[1,2,1,2] => [2,2,1,1]
=> 110110 => 6
[1,2,2,1] => [2,2,1,1]
=> 110110 => 6
[1,2,3] => [3,2,1]
=> 101010 => 6
[1,3,1,1] => [3,1,1,1]
=> 1001110 => 6
[1,3,2] => [3,2,1]
=> 101010 => 6
[1,4,1] => [4,1,1]
=> 1000110 => 6
[1,5] => [5,1]
=> 1000010 => 6
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => 6
[2,1,1,2] => [2,2,1,1]
=> 110110 => 6
[2,1,2,1] => [2,2,1,1]
=> 110110 => 6
Description
The number of inversions of a binary word.
Matching statistic: St000394
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> 1
[1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 5
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 5
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 5
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 5
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> 6
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> 6
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> 6
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> 6
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> 6
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> 6
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> 6
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> 6
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> 6
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 6
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000459
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> [1]
=> 1
[1,1] => [[1,1],[]]
=> [1,1]
=> 2
[2] => [[2],[]]
=> [2]
=> 2
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> 3
[1,2] => [[2,1],[]]
=> [2,1]
=> 3
[2,1] => [[2,2],[1]]
=> [2,2]
=> 3
[3] => [[3],[]]
=> [3]
=> 3
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> 4
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> 4
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 4
[1,3] => [[3,1],[]]
=> [3,1]
=> 4
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 4
[2,2] => [[3,2],[1]]
=> [3,2]
=> 4
[3,1] => [[3,3],[2]]
=> [3,3]
=> 4
[4] => [[4],[]]
=> [4]
=> 4
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 5
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> 5
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> 5
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> 5
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 5
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> 5
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> 5
[1,4] => [[4,1],[]]
=> [4,1]
=> 5
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 5
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> 5
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 5
[2,3] => [[4,2],[1]]
=> [4,2]
=> 5
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 5
[3,2] => [[4,3],[2]]
=> [4,3]
=> 5
[4,1] => [[4,4],[3]]
=> [4,4]
=> 5
[5] => [[5],[]]
=> [5]
=> 5
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> 6
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> 6
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> 6
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> 6
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> 6
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> 6
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> 6
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> 6
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> 6
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> 6
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> 6
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> 6
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> 6
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> 6
[1,5] => [[5,1],[]]
=> [5,1]
=> 6
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 6
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> 6
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> 6
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1
[1,1] => ([(0,1)],2)
=> [2]
=> 2
[2] => ([],2)
=> [1,1]
=> 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2] => ([(1,2)],3)
=> [2,1]
=> 3
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 3
[3] => ([],3)
=> [1,1,1]
=> 3
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,3] => ([(2,3)],4)
=> [2,1,1]
=> 4
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 4
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4
[4] => ([],4)
=> [1,1,1,1]
=> 4
[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)
=> [5]
=> 5
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 5
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 5
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 5
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 5
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 5
[5] => ([],5)
=> [1,1,1,1,1]
=> 5
[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)
=> [6]
=> 6
[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)
=> [5,1]
=> 6
[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)
=> [6]
=> 6
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 6
[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)
=> [6]
=> 6
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 6
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 6
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 6
[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)
=> [6]
=> 6
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 6
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 6
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 6
[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)
=> [6]
=> 6
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 6
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 6
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> 6
[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)
=> [6]
=> 6
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 6
[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)
=> [6]
=> 6
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000636
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[2] => ([],2)
=> ([],2)
=> 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 3
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3] => ([],3)
=> ([],3)
=> 3
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 4
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4] => ([],4)
=> ([],4)
=> 4
[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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 5
[1,2,1,1] => ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 5
[2,1,1,1] => ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 5
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5] => ([],5)
=> ([],5)
=> 5
[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),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[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)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[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)
=> ([(0,1),(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
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[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)
=> ([(0,1),(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
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,3,1] => ([(0,5),(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),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 6
[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)
=> ([(0,1),(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
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,2,2,1] => ([(0,5),(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),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[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)
=> ([(0,1),(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
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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
[1,5] => ([(4,5)],6)
=> ([(4,5)],6)
=> 6
[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)
=> ([(0,1),(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
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[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)
=> ([(0,1),(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
Description
The hull number of a graph.
The convex hull of a set of vertices $S$ of a graph is the smallest set $h(S)$ such that for any pair $u,v\in h(S)$ all vertices on a shortest path from $u$ to $v$ are also in $h(S)$.
The hull number is the size of the smallest set $S$ such that $h(S)$ is the set of all vertices.
Matching statistic: St000870
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1
[1,1] => ([(0,1)],2)
=> [2]
=> 2
[2] => ([],2)
=> [1,1]
=> 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2] => ([(1,2)],3)
=> [2,1]
=> 3
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 3
[3] => ([],3)
=> [1,1,1]
=> 3
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,3] => ([(2,3)],4)
=> [2,1,1]
=> 4
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 4
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4
[4] => ([],4)
=> [1,1,1,1]
=> 4
[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)
=> [5]
=> 5
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 5
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 5
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 5
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 5
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 5
[5] => ([],5)
=> [1,1,1,1,1]
=> 5
[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)
=> [6]
=> 6
[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)
=> [5,1]
=> 6
[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)
=> [6]
=> 6
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 6
[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)
=> [6]
=> 6
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 6
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 6
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 6
[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)
=> [6]
=> 6
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 6
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 6
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 6
[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)
=> [6]
=> 6
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 6
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 6
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> 6
[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)
=> [6]
=> 6
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 6
[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)
=> [6]
=> 6
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
The following 199 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001034The area of the parallelogram polyomino associated with the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000018The number of inversions of a permutation. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001437The flex of a binary word. St001523The degree of symmetry of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001672The restrained domination number of a graph. St000081The number of edges of a graph. St000921The number of internal inversions of a binary word. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001176The size of a partition minus its first part. St001479The number of bridges of a graph. St000867The sum of the hook lengths in the first row of an integer partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000806The semiperimeter of the associated bargraph. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000296The length of the symmetric border of a binary word. St000385The number of vertices with out-degree 1 in a binary tree. St000393The number of strictly increasing runs in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000627The exponent of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001267The length of the Lyndon factorization of the binary word. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000519The largest length of a factor maximising the subword complexity. St001342The number of vertices in the center of a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001759The Rajchgot index of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000719The number of alignments in a perfect matching. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000050The depth or height of a binary tree. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001430The number of positive entries in a signed permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000505The biggest entry in the block containing the 1. St000503The maximal difference between two elements in a common block. St000528The height of a poset. St001645The pebbling number of a connected graph. St000863The length of the first row of the shifted shape of a permutation. St001245The cyclic maximal difference between two consecutive entries of a permutation. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000553The number of blocks of a graph. St000625The sum of the minimal distances to a greater element. St000703The number of deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001052The length of the exterior of a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001096The size of the overlap set of a permutation. St000167The number of leaves of an ordered tree. St000451The length of the longest pattern of the form k 1 2. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000171The degree of the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001746The coalition number of a graph. St000019The cardinality of the support of a permutation. St000054The first entry of the permutation. St000060The greater neighbor of the maximum. St000619The number of cyclic descents of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001974The rank of the alternating sign matrix. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St001917The order of toric promotion on the set of labellings of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001622The number of join-irreducible elements of a lattice. St000672The number of minimal elements in Bruhat order not less than the permutation. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000080The rank of the poset. St000924The number of topologically connected components of a perfect matching. St001725The harmonious chromatic number of a graph. St000883The number of longest increasing subsequences of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001268The size of the largest ordinal summand in the poset. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000029The depth of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001480The number of simple summands of the module J^2/J^3. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001468The smallest fixpoint of a permutation. St000004The major index of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000653The last descent of a permutation. St000702The number of weak deficiencies of a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001428The number of B-inversions of a signed permutation. St001726The number of visible inversions of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000829The Ulam distance of a permutation to the identity permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001727The number of invisible inversions of a permutation. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000327The number of cover relations in a poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001875The number of simple modules with projective dimension at most 1. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000017The number of inversions of a standard tableau. St000820The number of compositions obtained by rotating the composition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001555The order of a signed permutation. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001927Sparre Andersen's number of positives of a signed permutation.
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