Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000708
Mapping
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 2
[1,1]
 => 1
[3]
 => 3
[2,1]
 => 2
[1,1,1]
 => 1
[4]
 => 4
[3,1]
 => 3
[2,2]
 => 4
[2,1,1]
 => 2
[1,1,1,1]
 => 1
[5]
 => 5
[4,1]
 => 4
[3,2]
 => 6
[3,1,1]
 => 3
[2,2,1]
 => 4
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 1
[6]
 => 6
[5,1]
 => 5
[4,2]
 => 8
[4,1,1]
 => 4
[3,3]
 => 9
[3,2,1]
 => 6
[3,1,1,1]
 => 3
[2,2,2]
 => 8
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 7
[6,1]
 => 6
[5,2]
 => 10
[5,1,1]
 => 5
[4,3]
 => 12
[4,2,1]
 => 8
[4,1,1,1]
 => 4
[3,3,1]
 => 9
[3,2,2]
 => 12
[3,2,1,1]
 => 6
[3,1,1,1,1]
 => 3
[2,2,2,1]
 => 8
[2,2,1,1,1]
 => 4
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1
[8]
 => 8
[7,1]
 => 7
[6,2]
 => 12
[6,1,1]
 => 6
[5,3]
 => 15
[5,2,1]
 => 10
[5,1,1,1]
 => 5
Description
The product of the parts of an integer partition.
Matching statistic: St000883
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Values
[2]
 => [[1,2]]
 => [1,2] => [2,1] => 2
[1,1]
 => [[1],[2]]
 => [2,1] => [1,2] => 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [3,2,1] => 3
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => [2,1,3] => 2
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 4
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => [3,2,1,4] => 3
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 4
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [2,1,3,4] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [5,4,3,2,1] => 5
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [4,3,2,1,5] => 4
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,2,1,5,4] => 6
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,2,1,4,5] => 3
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [2,1,4,3,5] => 4
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [2,1,3,4,5] => 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 5
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 8
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 9
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 6
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 8
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 4
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [2,1,3,4,5,6] => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 7
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 6
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 10
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 5
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ? = 12
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ? = 8
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 4
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => ? = 9
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ? = 12
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ? = 6
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => 3
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => ? = 8
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => [2,1,4,3,5,6,7] => ? = 4
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 8
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => 7
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 12
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 6
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => 15
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ? = 10
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => 5
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 16
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => ? = 12
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => 16
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => ? = 8
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => 4
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ? = 18
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ? = 9
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ? = 12
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ? = 6
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => 3
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 16
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ? = 8
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ? = 4
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => 2
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 1
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => 9
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8] => ? = 14
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7] => ? = 18
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9] => ? = 12
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6] => ? = 20
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9] => ? = 15
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8] => ? = 20
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9] => ? = 10
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9] => ? = 16
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8] => ? = 24
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8,9] => ? = 12
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,9] => ? = 16
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8,9] => ? = 8
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => [3,2,1,6,5,4,9,8,7] => ? = 27
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,9] => ? = 18
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8,9] => ? = 9
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,9,8] => ? = 24
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8,9] => ? = 12
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [9,8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8,9] => ? = 6
[2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,9] => ? = 16
[2,2,2,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8,9] => ? = 8
[2,2,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8,9] => ? = 4
[7,3]
 => [[1,2,3,4,5,6,7],[8,9,10]]
 => [8,9,10,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,10,9,8] => ? = 21
[7,2,1]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => [10,8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8,10] => ? = 14
[6,3,1]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => [10,7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7,10] => ? = 18
[6,2,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10]]
 => [10,9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9,10] => ? = 12
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [6,7,8,9,10,1,2,3,4,5] => [5,4,3,2,1,10,9,8,7,6] => ? = 25
[5,4,1]
 => [[1,2,3,4,5],[6,7,8,9],[10]]
 => [10,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,10] => ? = 20
[5,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10]]
 => [9,10,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,10,9] => ? = 30
[5,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10]]
 => [10,9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9,10] => ? = 15
[5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8,10] => ? = 20
[5,2,1,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9],[10]]
 => [10,9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9,10] => ? = 10
[4,4,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10]]
 => [10,9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9,10] => ? = 16
[4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => [8,9,10,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,10,9,8] => ? = 36
[4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8,10] => ? = 24
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001959
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001959: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 64%
Values
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 4
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 6
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 8
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 9
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 6
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 8
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 10
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 5
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 12
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 8
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 9
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 12
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 6
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 8
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 4
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 12
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 6
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 15
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 10
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 5
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 16
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 12
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 16
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 8
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 18
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 9
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 12
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 6
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 16
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 8
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 9
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 8
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 14
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 7
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 18
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => ? = 12
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 6
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 20
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => ? = 15
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 20
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => ? = 10
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 16
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 24
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 12
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 16
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 8
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 27
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 18
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 9
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 12
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 8
[2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 10
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 9
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => ? = 16
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 8
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 21
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
 => ? = 14
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 7
Description
The product of the heights of the peaks of a Dyck path.
Matching statistic: St000907
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 28%
Values
[2]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 2
[1,1]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 1
[3]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => 2
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([],3)
 => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 4
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 6
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 3
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 4
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 5
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => 8
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => 9
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => 6
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => 8
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => 4
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(4,5)],6)
 => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([],6)
 => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 6
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 10
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 5
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 12
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 8
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
 => 4
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 9
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 12
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 6
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
 => 3
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? = 8
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
 => 4
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(5,6)],7)
 => 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([],7)
 => 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 8
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 7
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
 => ? = 12
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
 => ? = 6
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
 => ? = 15
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
 => ? = 10
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
 => ? = 5
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => 16
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
 => ? = 12
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
 => ? = 16
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
 => ? = 8
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ([(4,5),(5,7),(7,6)],8)
 => ? = 4
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
 => ? = 18
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
 => 9
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
 => ? = 12
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
 => ? = 6
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
 => ? = 3
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => 16
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
 => ? = 8
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
 => 4
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => ([],8)
 => 1
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 9
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 8
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => ([(0,8),(1,3),(4,5),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 14
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [9,8,1,2,3,4,5,6,7] => ([(2,8),(4,5),(5,7),(6,4),(7,3),(8,6)],9)
 => ? = 7
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
 => ? = 18
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => ([(1,8),(2,4),(5,7),(6,5),(7,3),(8,6)],9)
 => ? = 12
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => ([(3,8),(5,7),(6,5),(7,4),(8,6)],9)
 => ? = 6
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
 => ? = 20
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
 => ? = 15
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => ([(0,5),(1,4),(2,8),(6,7),(7,3),(8,6)],9)
 => ? = 20
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => ([(2,8),(3,5),(6,7),(7,4),(8,6)],9)
 => ? = 10
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ([(4,5),(5,8),(7,6),(8,7)],9)
 => ? = 5
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ([(1,8),(2,7),(5,3),(6,4),(7,5),(8,6)],9)
 => ? = 16
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ([(0,5),(1,7),(2,8),(6,3),(7,4),(8,6)],9)
 => ? = 24
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ([(2,7),(3,8),(6,5),(7,6),(8,4)],9)
 => ? = 12
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => ([(1,6),(2,5),(3,8),(7,4),(8,7)],9)
 => ? = 16
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => ([(3,5),(4,6),(6,7),(7,8)],9)
 => ? = 8
[4,1,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,1,2,3,4] => ([(5,6),(6,8),(8,7)],9)
 => ? = 4
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => ([(0,8),(1,7),(2,6),(6,3),(7,4),(8,5)],9)
 => ? = 27
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => ([(1,6),(2,8),(3,7),(7,4),(8,5)],9)
 => ? = 18
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ([(3,6),(4,5),(5,7),(6,8)],9)
 => ? = 9
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => ([(0,7),(1,6),(2,5),(3,8),(8,4)],9)
 => ? = 24
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => ([(2,6),(3,5),(4,7),(7,8)],9)
 => ? = 12
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [9,8,7,6,4,5,1,2,3] => ([(4,6),(5,7),(7,8)],9)
 => ? = 6
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000911
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000911: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 28%
Values
[2]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 2
[1,1]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 1
[3]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => 2
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([],3)
 => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 4
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 6
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 3
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 4
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 5
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => 8
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => 9
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => 6
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => 8
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => 4
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(4,5)],6)
 => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([],6)
 => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 6
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 10
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 5
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 12
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 8
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
 => 4
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 9
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 12
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 6
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
 => 3
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? = 8
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
 => 4
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(5,6)],7)
 => 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([],7)
 => 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 8
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 7
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
 => ? = 12
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
 => ? = 6
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
 => ? = 15
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
 => ? = 10
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
 => ? = 5
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => 16
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
 => ? = 12
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
 => ? = 16
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
 => ? = 8
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ([(4,5),(5,7),(7,6)],8)
 => ? = 4
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
 => ? = 18
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
 => 9
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
 => ? = 12
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
 => ? = 6
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
 => ? = 3
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => 16
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
 => ? = 8
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
 => 4
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => ([],8)
 => 1
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 9
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 8
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => ([(0,8),(1,3),(4,5),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 14
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [9,8,1,2,3,4,5,6,7] => ([(2,8),(4,5),(5,7),(6,4),(7,3),(8,6)],9)
 => ? = 7
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
 => ? = 18
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => ([(1,8),(2,4),(5,7),(6,5),(7,3),(8,6)],9)
 => ? = 12
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => ([(3,8),(5,7),(6,5),(7,4),(8,6)],9)
 => ? = 6
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
 => ? = 20
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
 => ? = 15
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => ([(0,5),(1,4),(2,8),(6,7),(7,3),(8,6)],9)
 => ? = 20
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => ([(2,8),(3,5),(6,7),(7,4),(8,6)],9)
 => ? = 10
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ([(4,5),(5,8),(7,6),(8,7)],9)
 => ? = 5
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ([(1,8),(2,7),(5,3),(6,4),(7,5),(8,6)],9)
 => ? = 16
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ([(0,5),(1,7),(2,8),(6,3),(7,4),(8,6)],9)
 => ? = 24
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ([(2,7),(3,8),(6,5),(7,6),(8,4)],9)
 => ? = 12
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => ([(1,6),(2,5),(3,8),(7,4),(8,7)],9)
 => ? = 16
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => ([(3,5),(4,6),(6,7),(7,8)],9)
 => ? = 8
[4,1,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,1,2,3,4] => ([(5,6),(6,8),(8,7)],9)
 => ? = 4
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => ([(0,8),(1,7),(2,6),(6,3),(7,4),(8,5)],9)
 => ? = 27
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => ([(1,6),(2,8),(3,7),(7,4),(8,5)],9)
 => ? = 18
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ([(3,6),(4,5),(5,7),(6,8)],9)
 => ? = 9
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => ([(0,7),(1,6),(2,5),(3,8),(8,4)],9)
 => ? = 24
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => ([(2,6),(3,5),(4,7),(7,8)],9)
 => ? = 12
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [9,8,7,6,4,5,1,2,3] => ([(4,6),(5,7),(7,8)],9)
 => ? = 6
Description
The number of maximal antichains of maximal size in a poset.
Matching statistic: St000912
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 28%
Values
[2]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 2
[1,1]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 1
[3]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => 2
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([],3)
 => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 4
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 6
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 3
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 4
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 5
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => 8
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => 9
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => 6
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => 8
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => 4
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(4,5)],6)
 => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([],6)
 => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 6
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 10
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 5
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 12
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 8
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
 => 4
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 9
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 12
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 6
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
 => 3
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? = 8
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
 => 4
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(5,6)],7)
 => 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([],7)
 => 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 8
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 7
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
 => ? = 12
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
 => ? = 6
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
 => ? = 15
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
 => ? = 10
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
 => ? = 5
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => 16
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
 => ? = 12
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
 => ? = 16
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
 => ? = 8
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ([(4,5),(5,7),(7,6)],8)
 => ? = 4
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
 => ? = 18
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
 => 9
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
 => ? = 12
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
 => ? = 6
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
 => ? = 3
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => 16
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
 => ? = 8
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
 => 4
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => ([],8)
 => 1
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 9
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 8
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => ([(0,8),(1,3),(4,5),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 14
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [9,8,1,2,3,4,5,6,7] => ([(2,8),(4,5),(5,7),(6,4),(7,3),(8,6)],9)
 => ? = 7
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
 => ? = 18
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => ([(1,8),(2,4),(5,7),(6,5),(7,3),(8,6)],9)
 => ? = 12
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => ([(3,8),(5,7),(6,5),(7,4),(8,6)],9)
 => ? = 6
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
 => ? = 20
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
 => ? = 15
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => ([(0,5),(1,4),(2,8),(6,7),(7,3),(8,6)],9)
 => ? = 20
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => ([(2,8),(3,5),(6,7),(7,4),(8,6)],9)
 => ? = 10
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ([(4,5),(5,8),(7,6),(8,7)],9)
 => ? = 5
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ([(1,8),(2,7),(5,3),(6,4),(7,5),(8,6)],9)
 => ? = 16
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ([(0,5),(1,7),(2,8),(6,3),(7,4),(8,6)],9)
 => ? = 24
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ([(2,7),(3,8),(6,5),(7,6),(8,4)],9)
 => ? = 12
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => ([(1,6),(2,5),(3,8),(7,4),(8,7)],9)
 => ? = 16
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => ([(3,5),(4,6),(6,7),(7,8)],9)
 => ? = 8
[4,1,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,1,2,3,4] => ([(5,6),(6,8),(8,7)],9)
 => ? = 4
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => ([(0,8),(1,7),(2,6),(6,3),(7,4),(8,5)],9)
 => ? = 27
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => ([(1,6),(2,8),(3,7),(7,4),(8,5)],9)
 => ? = 18
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ([(3,6),(4,5),(5,7),(6,8)],9)
 => ? = 9
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => ([(0,7),(1,6),(2,5),(3,8),(8,4)],9)
 => ? = 24
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => ([(2,6),(3,5),(4,7),(7,8)],9)
 => ? = 12
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [9,8,7,6,4,5,1,2,3] => ([(4,6),(5,7),(7,8)],9)
 => ? = 6
Description
The number of maximal antichains in a poset.
Matching statistic: St001232
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 19%
Values
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 4 - 1
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[3,2]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 6 - 1
[3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 4 - 1
[2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[4,2]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 8 - 1
[4,1,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 9 - 1
[3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 6 - 1
[3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,2]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 8 - 1
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 4 - 1
[2,1,1,1,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 7 - 1
[6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[5,2]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 10 - 1
[5,1,1]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[4,3]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 12 - 1
[4,2,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 8 - 1
[4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[3,3,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 9 - 1
[3,2,2]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 12 - 1
[3,2,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => ? = 6 - 1
[3,1,1,1,1]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,2,1]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 8 - 1
[2,2,1,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 4 - 1
[2,1,1,1,1,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
[8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 8 - 1
[7,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 7 - 1
[6,2]
 => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 12 - 1
[6,1,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 6 - 1
[5,3]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 15 - 1
[5,2,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 10 - 1
[5,1,1,1]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => ? = 5 - 1
[4,4]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 16 - 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 12 - 1
[4,2,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 16 - 1
[4,2,1,1]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => ? = 8 - 1
[4,1,1,1,1]
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 4 - 1
[3,3,2]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 18 - 1
[3,3,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 9 - 1
[3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 12 - 1
[3,2,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
 => ? = 6 - 1
[3,1,1,1,1,1]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 3 - 1
[2,2,2,2]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 16 - 1
[2,2,2,1,1]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 8 - 1
[2,2,1,1,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 - 1
[2,1,1,1,1,1,1]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
[9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 9 - 1
[8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? = 8 - 1
[7,2]
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 14 - 1
[7,1,1]
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => ? = 7 - 1
[6,3]
 => [2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 18 - 1
[6,2,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 12 - 1
[6,1,1,1]
 => [4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => ? = 6 - 1
[5,4]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 20 - 1
[5,3,1]
 => [3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 15 - 1
[5,2,2]
 => [3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 20 - 1
[5,2,1,1]
 => [4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 10 - 1
[5,1,1,1,1]
 => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
 => ? = 5 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.