Your data matches 269 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000228
(load all 280 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3}}
 => [3]
 => 3
{{1,3},{2}}
 => [2,1]
 => 3
{{1},{2,3}}
 => [2,1]
 => 3
{{1},{2},{3}}
 => [1,1,1]
 => 3
{{1,2,3,4}}
 => [4]
 => 4
{{1,2,4},{3}}
 => [3,1]
 => 4
{{1,4},{2,3}}
 => [2,2]
 => 4
{{1},{2,3,4}}
 => [3,1]
 => 4
{{1},{2,4},{3}}
 => [2,1,1]
 => 4
{{1},{2},{3,4}}
 => [2,1,1]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => 4
{{1,2,3,4,5}}
 => [5]
 => 5
{{1,2,3,5},{4}}
 => [4,1]
 => 5
{{1,2,5},{3,4}}
 => [3,2]
 => 5
{{1,3,5},{2,4}}
 => [3,2]
 => 5
{{1},{2,3,4,5}}
 => [4,1]
 => 5
{{1},{2,3,5},{4}}
 => [3,1,1]
 => 5
{{1,5},{2,4},{3}}
 => [2,2,1]
 => 5
{{1},{2,5},{3,4}}
 => [2,2,1]
 => 5
{{1},{2},{3,4,5}}
 => [3,1,1]
 => 5
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => 5
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 5
{{1,2,3,4,5,6}}
 => [6]
 => 6
{{1,2,3,4,6},{5}}
 => [5,1]
 => 6
{{1,2,3,6},{4,5}}
 => [4,2]
 => 6
{{1,2,4,6},{3,5}}
 => [4,2]
 => 6
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3]
 => 6
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => 6
{{1},{2,3,4,5,6}}
 => [5,1]
 => 6
{{1},{2,3,4,6},{5}}
 => [4,1,1]
 => 6
{{1},{2,3,6},{4,5}}
 => [3,2,1]
 => 6
{{1},{2,4,6},{3,5}}
 => [3,2,1]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2]
 => 6
{{1},{2},{3,4,5,6}}
 => [4,1,1]
 => 6
{{1},{2},{3,4,6},{5}}
 => [3,1,1,1]
 => 6
{{1},{2,6},{3,5},{4}}
 => [2,2,1,1]
 => 6
{{1},{2},{3,6},{4,5}}
 => [2,2,1,1]
 => 6
{{1},{2},{3},{4,5,6}}
 => [3,1,1,1]
 => 6
{{1},{2},{3},{4,6},{5}}
 => [2,1,1,1,1]
 => 6
{{1},{2},{3},{4},{5,6}}
 => [2,1,1,1,1]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => 6
{{1,2,3,4,5,6,7}}
 => [7]
 => 7
{{1,2,3,4,5,7},{6}}
 => [6,1]
 => 7
{{1,2,3,4,7},{5,6}}
 => [5,2]
 => 7
{{1,2,3,5,7},{4,6}}
 => [5,2]
 => 7
{{1,2,3,7},{4,6},{5}}
 => [4,2,1]
 => 7
{{1,2,4,7},{3,5,6}}
 => [4,3]
 => 7
{{1,2,4,7},{3,6},{5}}
 => [4,2,1]
 => 7
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000293
(load all 17 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ 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%
Values
{{1,2,3}}
 => [3]
 => 1000 => 3
{{1,3},{2}}
 => [2,1]
 => 1010 => 3
{{1},{2,3}}
 => [2,1]
 => 1010 => 3
{{1},{2},{3}}
 => [1,1,1]
 => 1110 => 3
{{1,2,3,4}}
 => [4]
 => 10000 => 4
{{1,2,4},{3}}
 => [3,1]
 => 10010 => 4
{{1,4},{2,3}}
 => [2,2]
 => 1100 => 4
{{1},{2,3,4}}
 => [3,1]
 => 10010 => 4
{{1},{2,4},{3}}
 => [2,1,1]
 => 10110 => 4
{{1},{2},{3,4}}
 => [2,1,1]
 => 10110 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => 11110 => 4
{{1,2,3,4,5}}
 => [5]
 => 100000 => 5
{{1,2,3,5},{4}}
 => [4,1]
 => 100010 => 5
{{1,2,5},{3,4}}
 => [3,2]
 => 10100 => 5
{{1,3,5},{2,4}}
 => [3,2]
 => 10100 => 5
{{1},{2,3,4,5}}
 => [4,1]
 => 100010 => 5
{{1},{2,3,5},{4}}
 => [3,1,1]
 => 100110 => 5
{{1,5},{2,4},{3}}
 => [2,2,1]
 => 11010 => 5
{{1},{2,5},{3,4}}
 => [2,2,1]
 => 11010 => 5
{{1},{2},{3,4,5}}
 => [3,1,1]
 => 100110 => 5
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => 101110 => 5
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => 101110 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 111110 => 5
{{1,2,3,4,5,6}}
 => [6]
 => 1000000 => 6
{{1,2,3,4,6},{5}}
 => [5,1]
 => 1000010 => 6
{{1,2,3,6},{4,5}}
 => [4,2]
 => 100100 => 6
{{1,2,4,6},{3,5}}
 => [4,2]
 => 100100 => 6
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => 101010 => 6
{{1,3,6},{2,4,5}}
 => [3,3]
 => 11000 => 6
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => 101010 => 6
{{1},{2,3,4,5,6}}
 => [5,1]
 => 1000010 => 6
{{1},{2,3,4,6},{5}}
 => [4,1,1]
 => 1000110 => 6
{{1},{2,3,6},{4,5}}
 => [3,2,1]
 => 101010 => 6
{{1},{2,4,6},{3,5}}
 => [3,2,1]
 => 101010 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2]
 => 11100 => 6
{{1},{2},{3,4,5,6}}
 => [4,1,1]
 => 1000110 => 6
{{1},{2},{3,4,6},{5}}
 => [3,1,1,1]
 => 1001110 => 6
{{1},{2,6},{3,5},{4}}
 => [2,2,1,1]
 => 110110 => 6
{{1},{2},{3,6},{4,5}}
 => [2,2,1,1]
 => 110110 => 6
{{1},{2},{3},{4,5,6}}
 => [3,1,1,1]
 => 1001110 => 6
{{1},{2},{3},{4,6},{5}}
 => [2,1,1,1,1]
 => 1011110 => 6
{{1},{2},{3},{4},{5,6}}
 => [2,1,1,1,1]
 => 1011110 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => 1111110 => 6
{{1,2,3,4,5,6,7}}
 => [7]
 => 10000000 => 7
{{1,2,3,4,5,7},{6}}
 => [6,1]
 => 10000010 => 7
{{1,2,3,4,7},{5,6}}
 => [5,2]
 => 1000100 => 7
{{1,2,3,5,7},{4,6}}
 => [5,2]
 => 1000100 => 7
{{1,2,3,7},{4,6},{5}}
 => [4,2,1]
 => 1001010 => 7
{{1,2,4,7},{3,5,6}}
 => [4,3]
 => 101000 => 7
{{1,2,4,7},{3,6},{5}}
 => [4,2,1]
 => 1001010 => 7
Description
The number of inversions of a binary word.
Matching statistic: St000395
(load all 30 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
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%
Values
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 3
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 3
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 4
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 5
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 5
{{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 5
{{1},{2,3,5},{4}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
{{1,5},{2,4},{3}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5
{{1},{2,5},{3,4}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
{{1},{2},{3,4,5}}
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
{{1},{2},{3,5},{4}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
{{1,2,3,4,6},{5}}
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
{{1,2,3,6},{4,5}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 6
{{1,2,4,6},{3,5}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 6
{{1,2,6},{3,5},{4}}
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3,6},{2,5},{4}}
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
{{1},{2,3,4,5,6}}
 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6
{{1},{2,3,4,6},{5}}
 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 6
{{1},{2,3,6},{4,5}}
 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
{{1},{2,4,6},{3,5}}
 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1},{2},{3,4,5,6}}
 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 6
{{1},{2},{3,4,6},{5}}
 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 6
{{1},{2,6},{3,5},{4}}
 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 6
{{1},{2},{3,6},{4,5}}
 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 6
{{1},{2},{3},{4,5,6}}
 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 6
{{1},{2},{3},{4,6},{5}}
 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 6
{{1},{2},{3},{4},{5,6}}
 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
{{1,2,3,4,5,6,7}}
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7
{{1,2,3,4,5,7},{6}}
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 7
{{1,2,3,4,7},{5,6}}
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 7
{{1,2,3,5,7},{4,6}}
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 7
{{1,2,3,7},{4,6},{5}}
 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => 7
{{1,2,4,7},{3,5,6}}
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 7
{{1,2,4,7},{3,6},{5}}
 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => 7
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000459
(load all 17 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3}}
 => [2,3,1] => [2,1]
 => 3
{{1,3},{2}}
 => [3,2,1] => [3]
 => 3
{{1},{2,3}}
 => [1,3,2] => [2,1]
 => 3
{{1},{2},{3}}
 => [1,2,3] => [1,1,1]
 => 3
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,1]
 => 4
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1]
 => 4
{{1,4},{2,3}}
 => [4,3,2,1] => [4]
 => 4
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,1]
 => 4
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,1]
 => 4
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,1,1]
 => 4
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,1,1,1]
 => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,1,1,1]
 => 5
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [3,1,1]
 => 5
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,1]
 => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,1,1]
 => 5
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,1,1,1]
 => 5
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [3,1,1]
 => 5
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5]
 => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [4,1]
 => 5
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [2,1,1,1]
 => 5
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [3,1,1]
 => 5
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [2,1,1,1]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 5
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 6
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [3,1,1,1]
 => 6
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [4,1,1]
 => 6
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [3,1,1,1]
 => 6
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [5,1]
 => 6
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [4,1,1]
 => 6
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [4,1,1]
 => 6
{{1},{2,3,4,5,6}}
 => [1,3,4,5,6,2] => [2,1,1,1,1]
 => 6
{{1},{2,3,4,6},{5}}
 => [1,3,4,6,5,2] => [3,1,1,1]
 => 6
{{1},{2,3,6},{4,5}}
 => [1,3,6,5,4,2] => [4,1,1]
 => 6
{{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [3,1,1,1]
 => 6
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [6]
 => 6
{{1},{2},{3,4,5,6}}
 => [1,2,4,5,6,3] => [2,1,1,1,1]
 => 6
{{1},{2},{3,4,6},{5}}
 => [1,2,4,6,5,3] => [3,1,1,1]
 => 6
{{1},{2,6},{3,5},{4}}
 => [1,6,5,4,3,2] => [5,1]
 => 6
{{1},{2},{3,6},{4,5}}
 => [1,2,6,5,4,3] => [4,1,1]
 => 6
{{1},{2},{3},{4,5,6}}
 => [1,2,3,5,6,4] => [2,1,1,1,1]
 => 6
{{1},{2},{3},{4,6},{5}}
 => [1,2,3,6,5,4] => [3,1,1,1]
 => 6
{{1},{2},{3},{4},{5,6}}
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 6
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 7
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [3,1,1,1,1]
 => 7
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [4,1,1,1]
 => 7
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [3,1,1,1,1]
 => 7
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [5,1,1]
 => 7
{{1,2,4,7},{3,5,6}}
 => [2,4,5,7,6,3,1] => [4,1,1,1]
 => 7
{{1,2,4,7},{3,6},{5}}
 => [2,4,6,7,5,3,1] => [4,1,1,1]
 => 7
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 16 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3}}
 => [2,3,1] => [2,1]
 => 3
{{1,3},{2}}
 => [3,2,1] => [3]
 => 3
{{1},{2,3}}
 => [1,3,2] => [2,1]
 => 3
{{1},{2},{3}}
 => [1,2,3] => [1,1,1]
 => 3
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,1]
 => 4
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1]
 => 4
{{1,4},{2,3}}
 => [4,3,2,1] => [4]
 => 4
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,1]
 => 4
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,1]
 => 4
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,1,1]
 => 4
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,1,1,1]
 => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,1,1,1]
 => 5
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [3,1,1]
 => 5
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,1]
 => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,1,1]
 => 5
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,1,1,1]
 => 5
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [3,1,1]
 => 5
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5]
 => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [4,1]
 => 5
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [2,1,1,1]
 => 5
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [3,1,1]
 => 5
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [2,1,1,1]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 5
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 6
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [3,1,1,1]
 => 6
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [4,1,1]
 => 6
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [3,1,1,1]
 => 6
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [5,1]
 => 6
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [4,1,1]
 => 6
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [4,1,1]
 => 6
{{1},{2,3,4,5,6}}
 => [1,3,4,5,6,2] => [2,1,1,1,1]
 => 6
{{1},{2,3,4,6},{5}}
 => [1,3,4,6,5,2] => [3,1,1,1]
 => 6
{{1},{2,3,6},{4,5}}
 => [1,3,6,5,4,2] => [4,1,1]
 => 6
{{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [3,1,1,1]
 => 6
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [6]
 => 6
{{1},{2},{3,4,5,6}}
 => [1,2,4,5,6,3] => [2,1,1,1,1]
 => 6
{{1},{2},{3,4,6},{5}}
 => [1,2,4,6,5,3] => [3,1,1,1]
 => 6
{{1},{2,6},{3,5},{4}}
 => [1,6,5,4,3,2] => [5,1]
 => 6
{{1},{2},{3,6},{4,5}}
 => [1,2,6,5,4,3] => [4,1,1]
 => 6
{{1},{2},{3},{4,5,6}}
 => [1,2,3,5,6,4] => [2,1,1,1,1]
 => 6
{{1},{2},{3},{4,6},{5}}
 => [1,2,3,6,5,4] => [3,1,1,1]
 => 6
{{1},{2},{3},{4},{5,6}}
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 6
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 7
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [3,1,1,1,1]
 => 7
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [4,1,1,1]
 => 7
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [3,1,1,1,1]
 => 7
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [5,1,1]
 => 7
{{1,2,4,7},{3,5,6}}
 => [2,4,5,7,6,3,1] => [4,1,1,1]
 => 7
{{1,2,4,7},{3,6},{5}}
 => [2,4,6,7,5,3,1] => [4,1,1,1]
 => 7
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000725
(load all 16 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3}}
 => [2,3,1] => [1,3,2] => 3
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 3
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 3
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 3
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,4,3] => 4
{{1,2,4},{3}}
 => [2,4,3,1] => [1,4,3,2] => 4
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 4
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,4,3] => 4
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => 4
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 4
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,5,4] => 5
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,5,4,3] => 5
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,5,4,3,2] => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,2,5,4,3] => 5
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,2,3,5,4] => 5
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,2,5,4,3] => 5
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => 5
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,3,5,4] => 5
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [1,2,5,4,3] => 5
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => 5
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 6
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [1,2,3,6,5,4] => 6
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [1,2,6,5,4,3] => 6
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [1,2,3,6,5,4] => 6
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 6
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [1,2,6,5,4,3] => 6
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [1,2,6,5,4,3] => 6
{{1},{2,3,4,5,6}}
 => [1,3,4,5,6,2] => [1,2,3,4,6,5] => 6
{{1},{2,3,4,6},{5}}
 => [1,3,4,6,5,2] => [1,2,3,6,5,4] => 6
{{1},{2,3,6},{4,5}}
 => [1,3,6,5,4,2] => [1,2,6,5,4,3] => 6
{{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [1,2,3,6,5,4] => 6
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 6
{{1},{2},{3,4,5,6}}
 => [1,2,4,5,6,3] => [1,2,3,4,6,5] => 6
{{1},{2},{3,4,6},{5}}
 => [1,2,4,6,5,3] => [1,2,3,6,5,4] => 6
{{1},{2,6},{3,5},{4}}
 => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 6
{{1},{2},{3,6},{4,5}}
 => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 6
{{1},{2},{3},{4,5,6}}
 => [1,2,3,5,6,4] => [1,2,3,4,6,5] => 6
{{1},{2},{3},{4,6},{5}}
 => [1,2,3,6,5,4] => [1,2,3,6,5,4] => 6
{{1},{2},{3},{4},{5,6}}
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,7,6] => 7
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [1,2,3,4,7,6,5] => 7
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [1,2,3,7,6,5,4] => 7
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [1,2,3,4,7,6,5] => 7
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [1,2,7,6,5,4,3] => 7
{{1,2,4,7},{3,5,6}}
 => [2,4,5,7,6,3,1] => [1,2,3,7,6,5,4] => 7
{{1,2,4,7},{3,6},{5}}
 => [2,4,6,7,5,3,1] => [1,2,3,7,6,5,4] => 7
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000870
(load all 16 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3}}
 => [2,3,1] => [2,1]
 => 3
{{1,3},{2}}
 => [3,2,1] => [3]
 => 3
{{1},{2,3}}
 => [1,3,2] => [2,1]
 => 3
{{1},{2},{3}}
 => [1,2,3] => [1,1,1]
 => 3
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,1]
 => 4
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1]
 => 4
{{1,4},{2,3}}
 => [4,3,2,1] => [4]
 => 4
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,1]
 => 4
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,1]
 => 4
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,1,1]
 => 4
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,1,1,1]
 => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,1,1,1]
 => 5
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [3,1,1]
 => 5
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,1]
 => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,1,1]
 => 5
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,1,1,1]
 => 5
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [3,1,1]
 => 5
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5]
 => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [4,1]
 => 5
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [2,1,1,1]
 => 5
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [3,1,1]
 => 5
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [2,1,1,1]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 5
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 6
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [3,1,1,1]
 => 6
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [4,1,1]
 => 6
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [3,1,1,1]
 => 6
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [5,1]
 => 6
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [4,1,1]
 => 6
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [4,1,1]
 => 6
{{1},{2,3,4,5,6}}
 => [1,3,4,5,6,2] => [2,1,1,1,1]
 => 6
{{1},{2,3,4,6},{5}}
 => [1,3,4,6,5,2] => [3,1,1,1]
 => 6
{{1},{2,3,6},{4,5}}
 => [1,3,6,5,4,2] => [4,1,1]
 => 6
{{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [3,1,1,1]
 => 6
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [6]
 => 6
{{1},{2},{3,4,5,6}}
 => [1,2,4,5,6,3] => [2,1,1,1,1]
 => 6
{{1},{2},{3,4,6},{5}}
 => [1,2,4,6,5,3] => [3,1,1,1]
 => 6
{{1},{2,6},{3,5},{4}}
 => [1,6,5,4,3,2] => [5,1]
 => 6
{{1},{2},{3,6},{4,5}}
 => [1,2,6,5,4,3] => [4,1,1]
 => 6
{{1},{2},{3},{4,5,6}}
 => [1,2,3,5,6,4] => [2,1,1,1,1]
 => 6
{{1},{2},{3},{4,6},{5}}
 => [1,2,3,6,5,4] => [3,1,1,1]
 => 6
{{1},{2},{3},{4},{5,6}}
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 6
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 7
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [3,1,1,1,1]
 => 7
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [4,1,1,1]
 => 7
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [3,1,1,1,1]
 => 7
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [5,1,1]
 => 7
{{1,2,4,7},{3,5,6}}
 => [2,4,5,7,6,3,1] => [4,1,1,1]
 => 7
{{1,2,4,7},{3,6},{5}}
 => [2,4,6,7,5,3,1] => [4,1,1,1]
 => 7
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.
Matching statistic: St001004
(load all 51 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 3
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => 3
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 3
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 3
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 4
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => 4
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => 4
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => 4
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,2,3] => 4
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 4
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => 5
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => 5
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,2,3,4,5] => 5
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,3,4,5,2] => 5
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,3,5,2,4] => 5
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,3,4,5] => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,2,3,4] => 5
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,4,5,3] => 5
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [1,2,5,3,4] => 5
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => 5
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [1,2,3,4,6,5] => 6
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [1,2,3,6,4,5] => 6
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [1,2,4,5,6,3] => 6
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [1,2,6,3,4,5] => 6
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [1,2,3,4,6,5] => 6
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [1,2,3,5,6,4] => 6
{{1},{2,3,4,5,6}}
 => [1,3,4,5,6,2] => [1,3,4,5,6,2] => 6
{{1},{2,3,4,6},{5}}
 => [1,3,4,6,5,2] => [1,3,4,6,2,5] => 6
{{1},{2,3,6},{4,5}}
 => [1,3,6,5,4,2] => [1,3,6,2,4,5] => 6
{{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [1,4,5,6,2,3] => 6
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
{{1},{2},{3,4,5,6}}
 => [1,2,4,5,6,3] => [1,2,4,5,6,3] => 6
{{1},{2},{3,4,6},{5}}
 => [1,2,4,6,5,3] => [1,2,4,6,3,5] => 6
{{1},{2,6},{3,5},{4}}
 => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 6
{{1},{2},{3,6},{4,5}}
 => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 6
{{1},{2},{3},{4,5,6}}
 => [1,2,3,5,6,4] => [1,2,3,5,6,4] => 6
{{1},{2},{3},{4,6},{5}}
 => [1,2,3,6,5,4] => [1,2,3,6,4,5] => 6
{{1},{2},{3},{4},{5,6}}
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 7
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [1,2,3,4,5,7,6] => 7
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [1,2,3,4,7,5,6] => 7
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [1,2,3,5,6,7,4] => 7
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [1,2,3,7,4,5,6] => 7
{{1,2,4,7},{3,5,6}}
 => [2,4,5,7,6,3,1] => [1,2,4,5,7,3,6] => 7
{{1,2,4,7},{3,6},{5}}
 => [2,4,6,7,5,3,1] => [1,2,4,6,7,3,5] => 7
Description
The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St001020
(load all 32 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
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%
Values
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 3
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 3
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 4
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 5
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 5
{{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 5
{{1},{2,3,5},{4}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
{{1,5},{2,4},{3}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5
{{1},{2,5},{3,4}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
{{1},{2},{3,4,5}}
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
{{1},{2},{3,5},{4}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
{{1,2,3,4,6},{5}}
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
{{1,2,3,6},{4,5}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 6
{{1,2,4,6},{3,5}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 6
{{1,2,6},{3,5},{4}}
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3,6},{2,5},{4}}
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
{{1},{2,3,4,5,6}}
 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6
{{1},{2,3,4,6},{5}}
 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 6
{{1},{2,3,6},{4,5}}
 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
{{1},{2,4,6},{3,5}}
 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1},{2},{3,4,5,6}}
 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 6
{{1},{2},{3,4,6},{5}}
 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 6
{{1},{2,6},{3,5},{4}}
 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 6
{{1},{2},{3,6},{4,5}}
 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 6
{{1},{2},{3},{4,5,6}}
 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 6
{{1},{2},{3},{4,6},{5}}
 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 6
{{1},{2},{3},{4},{5,6}}
 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
{{1,2,3,4,5,6,7}}
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7
{{1,2,3,4,5,7},{6}}
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 7
{{1,2,3,4,7},{5,6}}
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 7
{{1,2,3,5,7},{4,6}}
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 7
{{1,2,3,7},{4,6},{5}}
 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => 7
{{1,2,4,7},{3,5,6}}
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 7
{{1,2,4,7},{3,6},{5}}
 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => 7
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001034
(load all 20 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3}}
 => [3]
 => [1,0,1,0,1,0]
 => 3
{{1,3},{2}}
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
{{1},{2,3}}
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
{{1},{2},{3}}
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
{{1,2,3,4}}
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,4},{3}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
{{1,4},{2,3}}
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
{{1},{2,3,4}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
{{1},{2,4},{3}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
{{1},{2},{3,4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 4
{{1,2,3,4,5}}
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,5},{4}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
{{1,2,5},{3,4}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
{{1,3,5},{2,4}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
{{1},{2,3,4,5}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 5
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 5
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
{{1,2,3,4,5,6}}
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 6
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6
{{1},{2,3,4,5,6}}
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 6
{{1},{2,3,4,6},{5}}
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 6
{{1},{2,3,6},{4,5}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6
{{1},{2,4,6},{3,5}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1},{2},{3,4,5,6}}
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 6
{{1},{2},{3,4,6},{5}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 6
{{1},{2,6},{3,5},{4}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 6
{{1},{2},{3,6},{4,5}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 6
{{1},{2},{3},{4,5,6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 6
{{1},{2},{3},{4,6},{5}}
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 6
{{1},{2},{3},{4},{5,6}}
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 6
{{1,2,3,4,5,6,7}}
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
{{1,2,3,4,5,7},{6}}
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 7
{{1,2,3,4,7},{5,6}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 7
{{1,2,3,5,7},{4,6}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 7
{{1,2,3,7},{4,6},{5}}
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 7
{{1,2,4,7},{3,5,6}}
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 7
{{1,2,4,7},{3,6},{5}}
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 7
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
The following 259 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001554The number of distinct nonempty subtrees of a binary tree. 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. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000806The semiperimeter of the associated bargraph. 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. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001267The length of the Lyndon factorization of the binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001437The flex of a binary word. St000060The greater neighbor of the maximum. St000018The number of inversions of a permutation. St000022The number of fixed points of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000203The number of external nodes of a binary tree. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000294The number of distinct factors of a binary word. St000308The height of the tree associated to a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000518The number of distinct subsequences in a binary word. St000528The height of a poset. St000548The number of different non-empty partial sums of an integer partition. St000625The sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000657The smallest part of an integer composition. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000808The number of up steps of the associated bargraph. St000863The length of the first row of the shifted shape of a permutation. St000873The aix statistic of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. 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. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001093The detour number of a graph. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. 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. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001523The degree of symmetry of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000050The depth or height of a binary tree. St000070The number of antichains in a poset. St000081The number of edges of a graph. St000189The number of elements in the poset. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St000296The length of the symmetric border of a binary word. St000441The number of successions of a permutation. St000520The number of patterns in a permutation. St000553The number of blocks of a graph. St000627The exponent of a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000672The number of minimal elements in Bruhat order not less than the permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000921The number of internal inversions of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001405The number of bonds in a permutation. 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. St001430The number of positive entries in a signed permutation. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001884The number of borders of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St000295The length of the border of a binary word. St000313The number of degree 2 vertices of a graph. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000519The largest length of a factor maximising the subword complexity. St000552The number of cut vertices of a graph. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001130The number of two successive successions in a permutation. St001308The number of induced paths on three vertices in a graph. St001368The number of vertices of maximal degree in a graph. St001521Half the total irregularity of a graph. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St000447The number of pairs of vertices of a graph with distance 3. St001306The number of induced paths on four vertices in a graph. St001759The Rajchgot index of a permutation. St000019The cardinality of the support of a permutation. St000653The last descent of a permutation. St001622The number of join-irreducible elements of a lattice. St000054The first entry of the permutation. St000141The maximum drop size of 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. St001958The degree of the polynomial interpolating the values of a permutation. 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. St000890The number of nonzero entries in an alternating sign matrix. St001925The minimal number of zeros in a row of an alternating sign matrix. 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. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000064The number of one-box pattern of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000338The number of pixed points of a permutation. St000501The size of the first part in the decomposition of a permutation. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000844The size of the largest block in the direct sum decomposition of a permutation. St000906The length of the shortest maximal chain in a poset. St000991The number of right-to-left minima of a 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. St001717The largest size of an interval in a poset. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000167The number of leaves of an ordered tree. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000316The number of non-left-to-right-maxima of a permutation. St000619The number of cyclic descents of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000956The maximal displacement of a permutation. 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. St001246The maximal difference between two consecutive entries of a permutation. St001391The disjunction number of a graph. St001516The number of cyclic bonds of a permutation. St001649The length of a longest trail in a graph. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000242The number of indices that are not cyclical small weak excedances. St000309The number of vertices with even degree. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001557The number of inversions of the second entry of a permutation. St000171The degree of the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St000924The number of topologically connected components of a perfect matching. St001556The number of inversions of the third entry of a permutation. St000029The depth of a permutation. 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. St001429The number of negative entries in a signed permutation. St001480The number of simple summands of the module J^2/J^3. St001511The minimal number of transpositions needed to sort a permutation in either direction. 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. St000327The number of cover relations in a poset. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000673The number of non-fixed points 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. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001927Sparre Andersen's number of positives of a signed permutation. St001948The number of augmented double ascents of a permutation. St001468The smallest fixpoint of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001520The number of strict 3-descents. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001645The pebbling number of a connected graph. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001434The number of negative sum pairs of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000135The number of lucky cars of the parking function. St000186The sum of the first row in a Gelfand-Tsetlin pattern. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001926Sparre Andersen's position of the maximum of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001845The number of join irreducibles minus the rank of a lattice. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree.