Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000048
Mapping
St000048: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 1
[1,1]
 => 2
[3]
 => 1
[2,1]
 => 3
[1,1,1]
 => 6
[4]
 => 1
[3,1]
 => 4
[2,2]
 => 6
[2,1,1]
 => 12
[1,1,1,1]
 => 24
[5]
 => 1
[4,1]
 => 5
[3,2]
 => 10
[3,1,1]
 => 20
[2,2,1]
 => 30
[2,1,1,1]
 => 60
[1,1,1,1,1]
 => 120
[6]
 => 1
[5,1]
 => 6
[4,2]
 => 15
[4,1,1]
 => 30
[3,3]
 => 20
[3,2,1]
 => 60
[3,1,1,1]
 => 120
[2,2,2]
 => 90
[2,2,1,1]
 => 180
[2,1,1,1,1]
 => 360
[1,1,1,1,1,1]
 => 720
[7]
 => 1
[6,1]
 => 7
[5,2]
 => 21
[5,1,1]
 => 42
[4,3]
 => 35
[4,2,1]
 => 105
[4,1,1,1]
 => 210
[3,3,1]
 => 140
[3,2,2]
 => 210
[3,2,1,1]
 => 420
[3,1,1,1,1]
 => 840
[2,2,2,1]
 => 630
[2,2,1,1,1]
 => 1260
[2,1,1,1,1,1]
 => 2520
[1,1,1,1,1,1,1]
 => 5040
[8]
 => 1
[7,1]
 => 8
[6,2]
 => 28
[6,1,1]
 => 56
[5,3]
 => 56
[5,2,1]
 => 168
Description
The multinomial of the parts of a partition. Given an integer partition $\lambda = [\lambda_1,\ldots,\lambda_k]$, this is the multinomial $$\binom{|\lambda|}{\lambda_1,\ldots,\lambda_k}.$$ For any integer composition $\mu$ that is a rearrangement of $\lambda$, this is the number of ordered set partitions whose list of block sizes is $\mu$.
Matching statistic: St000110
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 16%
Values
[1]
 => [[1]]
 => [1] => 1
[2]
 => [[1,2]]
 => [1,2] => 1
[1,1]
 => [[1],[2]]
 => [2,1] => 2
[3]
 => [[1,2,3]]
 => [1,2,3] => 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => 3
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 6
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 4
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 6
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 12
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 24
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 5
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 10
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 20
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 30
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 60
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 120
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 6
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => 15
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => 30
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 20
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => 60
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => 120
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 90
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 180
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => 360
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 720
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => 7
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => 21
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => 42
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => 35
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => 105
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => 210
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => 140
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => 210
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => 420
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => 840
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => 630
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => 1260
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => 2520
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => 5040
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => 1
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => 8
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => 28
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => 56
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => 56
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => 168
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => ? = 1
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => ? = 9
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => ? = 36
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [9,8,1,2,3,4,5,6,7] => ? = 72
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ? = 84
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => ? = 252
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => ? = 504
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ? = 126
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ? = 504
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => ? = 756
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => ? = 1512
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ? = 3024
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ? = 630
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ? = 1260
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ? = 2520
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => ? = 3780
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => ? = 7560
[4,1,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,1,2,3,4] => ? = 15120
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => ? = 1680
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => ? = 5040
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ? = 10080
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => ? = 7560
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => ? = 15120
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [9,8,7,6,4,5,1,2,3] => ? = 30240
[3,1,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,4,1,2,3] => ? = 60480
[2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => ? = 22680
[2,2,2,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,3,4,1,2] => ? = 45360
[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] => ? = 90720
[2,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,4,3,1,2] => ? = 181440
[1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,4,3,2,1] => ? = 362880
[10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [1,2,3,4,5,6,7,8,9,10] => ? = 1
[9,1]
 => [[1,2,3,4,5,6,7,8,9],[10]]
 => [10,1,2,3,4,5,6,7,8,9] => ? = 10
[8,2]
 => [[1,2,3,4,5,6,7,8],[9,10]]
 => [9,10,1,2,3,4,5,6,7,8] => ? = 45
[8,1,1]
 => [[1,2,3,4,5,6,7,8],[9],[10]]
 => [10,9,1,2,3,4,5,6,7,8] => ? = 90
[7,3]
 => [[1,2,3,4,5,6,7],[8,9,10]]
 => [8,9,10,1,2,3,4,5,6,7] => ? = 120
[7,2,1]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => [10,8,9,1,2,3,4,5,6,7] => ? = 360
[7,1,1,1]
 => [[1,2,3,4,5,6,7],[8],[9],[10]]
 => [10,9,8,1,2,3,4,5,6,7] => ? = 720
[6,4]
 => [[1,2,3,4,5,6],[7,8,9,10]]
 => [7,8,9,10,1,2,3,4,5,6] => ? = 210
[6,3,1]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => [10,7,8,9,1,2,3,4,5,6] => ? = 840
[6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => [9,10,7,8,1,2,3,4,5,6] => ? = 1260
[6,2,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10]]
 => [10,9,7,8,1,2,3,4,5,6] => ? = 2520
[6,1,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9],[10]]
 => [10,9,8,7,1,2,3,4,5,6] => ? = 5040
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [6,7,8,9,10,1,2,3,4,5] => ? = 252
[5,4,1]
 => [[1,2,3,4,5],[6,7,8,9],[10]]
 => [10,6,7,8,9,1,2,3,4,5] => ? = 1260
[5,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10]]
 => [9,10,6,7,8,1,2,3,4,5] => ? = 2520
[5,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10]]
 => [10,9,6,7,8,1,2,3,4,5] => ? = 5040
[5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,1,2,3,4,5] => ? = 7560
[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] => ? = 15120
[5,1,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9],[10]]
 => [10,9,8,7,6,1,2,3,4,5] => ? = 30240
[4,4,2]
 => [[1,2,3,4],[5,6,7,8],[9,10]]
 => [9,10,5,6,7,8,1,2,3,4] => ? = 3150
Description
The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000100
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 8%
Values
[1]
 => [[1]]
 => [1] => ([],1)
 => ? = 1
[2]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 1
[1,1]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 2
[3]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => 3
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([],3)
 => 6
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 4
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 6
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => 12
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => 24
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 5
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 10
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 20
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 30
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 60
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => 120
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 6
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => 15
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => 30
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => 20
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => 60
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => 120
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => 90
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => 180
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(4,5)],6)
 => 360
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([],6)
 => 720
[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)
 => 1
[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)
 => ? = 7
[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)
 => ? = 21
[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)
 => ? = 42
[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)
 => ? = 35
[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)
 => ? = 105
[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)
 => ? = 210
[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)
 => ? = 140
[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)
 => ? = 210
[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)
 => ? = 420
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
 => ? = 840
[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)
 => ? = 630
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
 => ? = 1260
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(5,6)],7)
 => ? = 2520
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([],7)
 => ? = 5040
[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)
 => 1
[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)
 => ? = 8
[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)
 => ? = 28
[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)
 => ? = 56
[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)
 => ? = 56
[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)
 => ? = 168
[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)
 => ? = 336
[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)
 => 70
[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)
 => ? = 280
[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)
 => ? = 420
[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)
 => ? = 840
[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)
 => ? = 1680
[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)
 => ? = 560
[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)
 => 1120
[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)
 => ? = 1680
[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)
 => ? = 3360
[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)
 => ? = 6720
[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)
 => 2520
[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)
 => ? = 5040
[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)
 => 10080
[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)
 => ? = 20160
[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)
 => 40320
[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)
 => 1
[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)
 => ? = 9
[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)
 => ? = 36
[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)
 => ? = 72
[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)
 => ? = 84
[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)
 => ? = 252
[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)
 => ? = 504
[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)
 => ? = 126
[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)
 => ? = 504
[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)
 => ? = 756
[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)
 => ? = 1512
[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)
 => ? = 3024
[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)
 => ? = 630
[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)
 => ? = 1260
[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)
 => ? = 2520
[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)
 => ? = 3780
[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)
 => ? = 7560
[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)
 => ? = 15120
[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)
 => ? = 1680
[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)
 => ? = 5040
[10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [1,2,3,4,5,6,7,8,9,10] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => 1
Description
The number of linear extensions of a poset.