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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000100
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => ([],2)
=> 2
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 4
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> 6
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 8
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 8
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 5
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 10
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> 12
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 12
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> 12
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 10
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> 12
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 15
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 15
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 10
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> 24
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 15
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 10
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 15
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> 12
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 18
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 18
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 20
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 30
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 20
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 20
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 20
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 30
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 18
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 20
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 18
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> 24
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6)
=> 36
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> 24
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> 15
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 40
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 30
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 40
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> 24
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 30
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 30
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 40
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => ([(1,5),(2,5),(3,5),(5,4)],6)
=> 36
Description
The number of linear extensions of a poset.
Matching statistic: St000110
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 2
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 6
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 4
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 8
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 6
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 8
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 5
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 10
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 12
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 12
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 12
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 10
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 12
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 15
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 15
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 10
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 24
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 15
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 10
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 15
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 12
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 18
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 18
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 20
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 30
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 20
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 20
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 20
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 30
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 18
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 20
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 18
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 24
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 36
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 24
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 15
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 40
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 30
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 40
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => 24
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 30
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 30
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 40
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => 36
[6,5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,3,2,4,1] => ? = 4480
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: St000063
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000063: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 92%●distinct values known / distinct values provided: 87%
Values
[1]
=> 2
[2]
=> 3
[1,1]
=> 3
[3]
=> 4
[2,1]
=> 6
[1,1,1]
=> 4
[4]
=> 5
[3,1]
=> 8
[2,2]
=> 6
[2,1,1]
=> 8
[1,1,1,1]
=> 5
[5]
=> 6
[4,1]
=> 10
[3,2]
=> 12
[3,1,1]
=> 12
[2,2,1]
=> 12
[2,1,1,1]
=> 10
[1,1,1,1,1]
=> 6
[5,1]
=> 12
[4,2]
=> 15
[4,1,1]
=> 15
[3,3]
=> 10
[3,2,1]
=> 24
[3,1,1,1]
=> 15
[2,2,2]
=> 10
[2,2,1,1]
=> 15
[2,1,1,1,1]
=> 12
[5,2]
=> 18
[5,1,1]
=> 18
[4,3]
=> 20
[4,2,1]
=> 30
[4,1,1,1]
=> 20
[3,3,1]
=> 20
[3,2,2]
=> 20
[3,2,1,1]
=> 30
[3,1,1,1,1]
=> 18
[2,2,2,1]
=> 20
[2,2,1,1,1]
=> 18
[5,3]
=> 24
[5,2,1]
=> 36
[5,1,1,1]
=> 24
[4,4]
=> 15
[4,3,1]
=> 40
[4,2,2]
=> 30
[4,2,1,1]
=> 40
[4,1,1,1,1]
=> 24
[3,3,2]
=> 30
[3,3,1,1]
=> 30
[3,2,2,1]
=> 40
[3,2,1,1,1]
=> 36
[4,4,4,4]
=> ? = 70
[7,6,5,4,3,2,1]
=> ? = 40320
[6,6,4,4,3,2,1]
=> ? = 10080
[6,6,5,3,3,2,1]
=> ? = 10080
[6,6,5,4,2,2,1]
=> ? = 10080
[6,6,5,4,3,1,1]
=> ? = 10080
[6,5,5,4,2,1,1]
=> ? = 4480
[6,6,5,4,3,2]
=> ? = 10080
[7,5,5,4,3,2]
=> ? = 10080
[7,6,4,4,3,2]
=> ? = 10080
[6,5,5,4,3,1]
=> ? = 4480
Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St000085
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000085: Ordered trees ⟶ ℤResult quality: 87% ●values known / values provided: 92%●distinct values known / distinct values provided: 87%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000085: Ordered trees ⟶ ℤResult quality: 87% ●values known / values provided: 92%●distinct values known / distinct values provided: 87%
Values
[1]
=> [1,0,1,0]
=> [[],[]]
=> 2
[2]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 4
[2,1]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 6
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 8
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 6
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 8
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 5
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 10
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 12
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 12
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 12
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 10
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[[],[]]]],[]]
=> 12
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[[]],[]],[]]
=> 15
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 15
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 10
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 24
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 15
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 10
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 15
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[],[[[[],[]]]]]
=> 12
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[[[]],[]]],[]]
=> 18
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[[],[[]]]],[]]
=> 18
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 20
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 30
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 20
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 20
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 20
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 30
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[],[[[[]],[]]]]
=> 18
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 20
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[],[[[],[[]]]]]
=> 18
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[[[]]],[]],[]]
=> 24
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[[[],[],[]]],[]]
=> 36
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[[],[[[]]]],[]]
=> 24
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[[]]]],[[]]]
=> 15
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 40
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 30
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 40
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[],[[[[]]],[]]]
=> 24
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 30
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 30
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 40
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[],[[[],[],[]]]]
=> 36
[4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 70
[7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[]]
=> ? = 40320
[6,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[],[],[[]],[[]]]
=> ? = 10080
[6,6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[],[],[],[[]],[],[[]]]
=> ? = 10080
[6,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[],[],[[]],[],[],[[]]]
=> ? = 10080
[6,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[[]],[],[],[],[[]]]
=> ? = 10080
[6,5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [[],[[],[]],[],[[],[]]]
=> ? = 4480
[6,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[]],[],[],[],[],[[]]]
=> ? = 10080
[7,5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[]],[],[],[],[[]],[]]
=> ? = 10080
[7,6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[]],[],[],[[]],[],[]]
=> ? = 10080
[6,5,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [[[],[]],[],[],[[],[]]]
=> ? = 4480
Description
The number of linear extensions of the tree.
We use Knuth's hook length formula for trees [pg.70, 1]. For an ordered tree $T$ on $n$ vertices, the number of linear extensions is
$$
\frac{n!}{\prod_{v\in T}|T_v|},
$$
where $T_v$ is the number of vertices of the subtree rooted at $v$.
Matching statistic: St000071
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000071: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 26%
Mp00262: Binary words —poset of factors⟶ Posets
St000071: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 26%
Values
[1]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[1,1]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[3]
=> 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4
[2,1]
=> 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 6
[1,1,1]
=> 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4
[4]
=> 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 5
[3,1]
=> 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 8
[2,2]
=> 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 6
[2,1,1]
=> 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 8
[1,1,1,1]
=> 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 5
[5]
=> 100000 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> 6
[4,1]
=> 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 10
[3,2]
=> 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 12
[3,1,1]
=> 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? = 12
[2,2,1]
=> 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 12
[2,1,1,1]
=> 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 10
[1,1,1,1,1]
=> 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> 6
[5,1]
=> 1000010 => ([(0,4),(0,5),(1,3),(1,9),(1,17),(2,14),(2,19),(3,2),(3,13),(3,18),(4,15),(4,16),(5,1),(5,15),(5,16),(7,11),(8,7),(9,13),(10,8),(11,6),(12,6),(13,14),(14,12),(15,9),(15,10),(16,10),(16,17),(17,8),(17,18),(18,7),(18,19),(19,11),(19,12)],20)
=> ? = 12
[4,2]
=> 100100 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 15
[4,1,1]
=> 1000110 => ([(0,4),(0,5),(1,13),(1,20),(2,3),(2,14),(2,21),(3,8),(3,16),(4,1),(4,17),(4,18),(5,2),(5,17),(5,18),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,19),(16,10),(16,19),(17,20),(17,21),(18,13),(18,14),(19,11),(19,12),(20,7),(20,15),(21,15),(21,16)],22)
=> ? = 15
[3,3]
=> 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 10
[3,2,1]
=> 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 24
[3,1,1,1]
=> 1001110 => ([(0,4),(0,5),(1,13),(1,20),(2,3),(2,14),(2,21),(3,8),(3,16),(4,1),(4,17),(4,18),(5,2),(5,17),(5,18),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,19),(16,10),(16,19),(17,20),(17,21),(18,13),(18,14),(19,11),(19,12),(20,7),(20,15),(21,15),(21,16)],22)
=> ? = 15
[2,2,2]
=> 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 10
[2,2,1,1]
=> 110110 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 15
[2,1,1,1,1]
=> 1011110 => ([(0,4),(0,5),(1,3),(1,9),(1,17),(2,14),(2,19),(3,2),(3,13),(3,18),(4,15),(4,16),(5,1),(5,15),(5,16),(7,11),(8,7),(9,13),(10,8),(11,6),(12,6),(13,14),(14,12),(15,9),(15,10),(16,10),(16,17),(17,8),(17,18),(18,7),(18,19),(19,11),(19,12)],20)
=> ? = 12
[5,2]
=> 1000100 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
=> ? = 18
[5,1,1]
=> 10000110 => ([(0,5),(0,6),(1,4),(1,16),(1,26),(2,17),(2,25),(3,10),(3,22),(4,3),(4,8),(4,18),(5,1),(5,20),(5,21),(6,2),(6,20),(6,21),(8,10),(9,11),(10,13),(11,12),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,22),(18,23),(19,11),(19,23),(20,25),(20,26),(21,16),(21,17),(22,13),(22,24),(23,12),(23,24),(24,14),(24,15),(25,9),(25,19),(26,18),(26,19)],27)
=> ? = 18
[4,3]
=> 101000 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 20
[4,2,1]
=> 1001010 => ([(0,2),(0,3),(1,5),(1,12),(2,18),(2,19),(3,1),(3,18),(3,19),(5,6),(6,7),(7,10),(8,11),(9,8),(10,4),(11,4),(12,6),(12,14),(13,9),(13,15),(14,7),(14,16),(15,8),(15,16),(16,10),(16,11),(17,9),(17,14),(17,15),(18,5),(18,13),(18,17),(19,12),(19,13),(19,17)],20)
=> ? = 30
[4,1,1,1]
=> 10001110 => ([(0,5),(0,6),(1,4),(1,17),(1,27),(2,3),(2,16),(2,26),(3,8),(3,19),(4,9),(4,20),(5,2),(5,21),(5,22),(6,1),(6,21),(6,22),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,23),(18,24),(19,10),(19,23),(20,11),(20,24),(21,26),(21,27),(22,16),(22,17),(23,12),(23,25),(24,13),(24,25),(25,14),(25,15),(26,18),(26,19),(27,18),(27,20)],28)
=> ? = 20
[3,3,1]
=> 110010 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 20
[3,2,2]
=> 101100 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 20
[3,2,1,1]
=> 1010110 => ([(0,2),(0,3),(1,5),(1,12),(2,18),(2,19),(3,1),(3,18),(3,19),(5,6),(6,7),(7,10),(8,11),(9,8),(10,4),(11,4),(12,6),(12,14),(13,9),(13,15),(14,7),(14,16),(15,8),(15,16),(16,10),(16,11),(17,9),(17,14),(17,15),(18,5),(18,13),(18,17),(19,12),(19,13),(19,17)],20)
=> ? = 30
[3,1,1,1,1]
=> 10011110 => ([(0,5),(0,6),(1,4),(1,16),(1,26),(2,17),(2,25),(3,10),(3,22),(4,3),(4,8),(4,18),(5,1),(5,20),(5,21),(6,2),(6,20),(6,21),(8,10),(9,11),(10,13),(11,12),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,22),(18,23),(19,11),(19,23),(20,25),(20,26),(21,16),(21,17),(22,13),(22,24),(23,12),(23,24),(24,14),(24,15),(25,9),(25,19),(26,18),(26,19)],27)
=> ? = 18
[2,2,2,1]
=> 111010 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 20
[2,2,1,1,1]
=> 1101110 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
=> ? = 18
[5,3]
=> 1001000 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
=> ? = 24
[5,2,1]
=> 10001010 => ([(0,3),(0,4),(1,2),(1,14),(1,22),(2,6),(2,15),(3,23),(3,24),(4,1),(4,23),(4,24),(6,8),(7,9),(8,10),(9,13),(10,12),(11,7),(12,5),(13,5),(14,6),(15,8),(15,19),(16,19),(16,20),(17,11),(17,18),(18,7),(18,20),(19,10),(19,21),(20,9),(20,21),(21,12),(21,13),(22,15),(22,16),(23,17),(23,22),(23,25),(24,14),(24,17),(24,25),(25,11),(25,16),(25,18)],26)
=> ? = 36
[5,1,1,1]
=> 100001110 => ([(0,6),(0,7),(1,4),(1,11),(1,28),(2,5),(2,12),(2,27),(3,18),(3,33),(4,20),(4,32),(5,3),(5,19),(5,31),(6,1),(6,29),(6,30),(7,2),(7,29),(7,30),(9,13),(10,14),(11,20),(12,19),(13,15),(14,16),(15,17),(16,8),(17,8),(18,10),(19,18),(20,9),(21,22),(21,25),(22,23),(22,26),(23,15),(23,24),(24,16),(24,17),(25,13),(25,23),(26,14),(26,24),(27,21),(27,31),(28,21),(28,32),(29,27),(29,28),(30,11),(30,12),(31,22),(31,33),(32,9),(32,25),(33,10),(33,26)],34)
=> ? = 24
[4,4]
=> 110000 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> 15
[4,3,1]
=> 1010010 => ([(0,2),(0,3),(1,12),(1,13),(2,18),(2,19),(3,1),(3,18),(3,19),(5,8),(6,5),(7,10),(8,11),(9,7),(10,4),(11,4),(12,9),(12,15),(13,14),(13,15),(14,8),(14,16),(15,7),(15,16),(16,10),(16,11),(17,5),(17,9),(17,14),(18,6),(18,12),(18,17),(19,6),(19,13),(19,17)],20)
=> ? = 40
[4,2,2]
=> 1001100 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ? = 30
[4,2,1,1]
=> 10010110 => ([(0,3),(0,4),(1,15),(1,25),(2,14),(2,24),(3,2),(3,26),(3,27),(4,1),(4,26),(4,27),(6,8),(7,9),(8,10),(9,11),(10,12),(11,13),(12,5),(13,5),(14,6),(15,7),(16,18),(16,23),(17,19),(17,23),(18,8),(18,21),(19,9),(19,22),(20,12),(20,13),(21,10),(21,20),(22,11),(22,20),(23,21),(23,22),(24,6),(24,18),(25,7),(25,19),(26,16),(26,17),(26,24),(26,25),(27,14),(27,15),(27,16),(27,17)],28)
=> ? = 40
[4,1,1,1,1]
=> 100011110 => ([(0,6),(0,7),(1,4),(1,11),(1,28),(2,5),(2,12),(2,27),(3,18),(3,33),(4,20),(4,32),(5,3),(5,19),(5,31),(6,1),(6,29),(6,30),(7,2),(7,29),(7,30),(9,13),(10,14),(11,20),(12,19),(13,15),(14,16),(15,17),(16,8),(17,8),(18,10),(19,18),(20,9),(21,22),(21,25),(22,23),(22,26),(23,15),(23,24),(24,16),(24,17),(25,13),(25,23),(26,14),(26,24),(27,21),(27,31),(28,21),(28,32),(29,27),(29,28),(30,11),(30,12),(31,22),(31,33),(32,9),(32,25),(33,10),(33,26)],34)
=> ? = 24
[3,3,2]
=> 110100 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> ? = 30
[3,3,1,1]
=> 1100110 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ? = 30
[3,2,2,1]
=> 1011010 => ([(0,2),(0,3),(1,12),(1,13),(2,18),(2,19),(3,1),(3,18),(3,19),(5,8),(6,5),(7,10),(8,11),(9,7),(10,4),(11,4),(12,9),(12,15),(13,14),(13,15),(14,8),(14,16),(15,7),(15,16),(16,10),(16,11),(17,5),(17,9),(17,14),(18,6),(18,12),(18,17),(19,6),(19,13),(19,17)],20)
=> ? = 40
[3,2,1,1,1]
=> 10101110 => ([(0,3),(0,4),(1,2),(1,14),(1,22),(2,6),(2,15),(3,23),(3,24),(4,1),(4,23),(4,24),(6,8),(7,9),(8,10),(9,13),(10,12),(11,7),(12,5),(13,5),(14,6),(15,8),(15,19),(16,19),(16,20),(17,11),(17,18),(18,7),(18,20),(19,10),(19,21),(20,9),(20,21),(21,12),(21,13),(22,15),(22,16),(23,17),(23,22),(23,25),(24,14),(24,17),(24,25),(25,11),(25,16),(25,18)],26)
=> ? = 36
[2,2,2,2]
=> 111100 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> 15
[2,2,2,1,1]
=> 1110110 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
=> ? = 24
[5,4]
=> 1010000 => ([(0,4),(0,5),(1,3),(1,14),(2,12),(3,2),(3,18),(4,17),(4,19),(5,1),(5,17),(5,19),(7,8),(8,9),(9,10),(10,6),(11,6),(12,11),(13,8),(13,16),(14,16),(14,18),(15,10),(15,11),(16,9),(16,15),(17,7),(17,13),(18,12),(18,15),(19,7),(19,13),(19,14)],20)
=> ? = 30
[5,3,1]
=> 10010010 => ([(0,2),(0,3),(1,14),(1,15),(2,19),(2,20),(3,1),(3,19),(3,20),(5,7),(6,8),(7,4),(8,4),(9,16),(9,18),(10,17),(10,18),(11,16),(11,17),(12,7),(12,8),(13,9),(13,11),(14,9),(14,10),(15,10),(15,11),(16,5),(16,12),(17,6),(17,12),(18,5),(18,6),(19,13),(19,14),(20,13),(20,15)],21)
=> ? = 48
[5,2,2]
=> 10001100 => ([(0,4),(0,5),(1,23),(1,25),(2,3),(2,24),(2,26),(3,8),(3,18),(4,1),(4,19),(4,20),(5,2),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,10),(12,9),(13,6),(14,6),(15,13),(15,14),(16,12),(16,21),(17,21),(17,22),(18,11),(18,22),(19,23),(19,24),(20,25),(20,26),(21,9),(21,15),(22,10),(22,15),(23,16),(23,17),(24,17),(24,18),(25,7),(25,16),(26,7),(26,8)],27)
=> ? = 36
[5,2,1,1]
=> 100010110 => ([(0,4),(0,5),(1,18),(1,32),(2,3),(2,17),(2,31),(3,8),(3,30),(4,1),(4,33),(4,34),(5,2),(5,33),(5,34),(7,14),(8,11),(9,15),(10,16),(11,12),(12,9),(13,10),(14,13),(15,6),(16,6),(17,8),(18,7),(19,21),(19,22),(20,19),(20,27),(21,10),(21,23),(22,9),(22,23),(23,15),(23,16),(24,19),(24,28),(25,20),(25,29),(26,20),(26,24),(27,13),(27,21),(28,12),(28,22),(29,14),(29,27),(30,11),(30,28),(31,24),(31,30),(32,7),(32,29),(33,25),(33,26),(33,31),(33,32),(34,17),(34,18),(34,25),(34,26)],35)
=> ? = 48
[5,1,1,1,1]
=> 1000011110 => ([(0,7),(0,8),(1,6),(1,13),(1,34),(2,5),(2,12),(2,33),(3,20),(3,39),(4,21),(4,40),(5,3),(5,22),(5,37),(6,4),(6,23),(6,38),(7,2),(7,35),(7,36),(8,1),(8,35),(8,36),(10,14),(11,15),(12,22),(13,23),(14,16),(15,17),(16,18),(17,19),(18,9),(19,9),(20,10),(21,11),(22,20),(23,21),(24,16),(24,30),(25,17),(25,30),(26,24),(26,25),(27,31),(27,32),(28,14),(28,24),(29,15),(29,25),(30,18),(30,19),(31,26),(31,28),(32,26),(32,29),(33,27),(33,37),(34,27),(34,38),(35,33),(35,34),(36,12),(36,13),(37,31),(37,39),(38,32),(38,40),(39,10),(39,28),(40,11),(40,29)],41)
=> ? = 30
[4,4,1]
=> 1100010 => ([(0,4),(0,5),(1,14),(2,3),(2,19),(2,20),(3,15),(3,16),(4,1),(4,17),(4,21),(5,2),(5,17),(5,21),(7,10),(8,9),(9,12),(10,13),(11,7),(12,6),(13,6),(14,8),(15,10),(15,18),(16,9),(16,18),(17,11),(17,20),(18,12),(18,13),(19,8),(19,16),(20,7),(20,15),(21,11),(21,14),(21,19)],22)
=> ? = 30
[4,3,2]
=> 1010100 => ([(0,2),(0,3),(1,10),(2,14),(2,17),(3,1),(3,14),(3,17),(5,8),(6,5),(7,9),(8,4),(9,4),(10,6),(11,13),(11,16),(12,8),(12,9),(13,7),(13,12),(14,11),(14,15),(15,6),(15,13),(15,16),(16,5),(16,7),(16,12),(17,10),(17,11),(17,15)],18)
=> ? = 60
[4,3,1,1]
=> 10100110 => ([(0,3),(0,4),(1,23),(1,25),(2,15),(2,24),(3,1),(3,26),(3,27),(4,2),(4,26),(4,27),(6,10),(7,8),(8,9),(9,11),(10,12),(11,14),(12,13),(13,5),(14,5),(15,6),(16,20),(16,21),(17,10),(17,20),(18,8),(18,19),(19,9),(19,21),(20,12),(20,22),(21,11),(21,22),(22,13),(22,14),(23,16),(23,19),(24,6),(24,17),(25,16),(25,17),(26,7),(26,18),(26,24),(26,25),(27,7),(27,15),(27,18),(27,23)],28)
=> ? = 60
[4,2,2,1]
=> 10011010 => ([(0,3),(0,4),(1,23),(1,25),(2,15),(2,24),(3,1),(3,26),(3,27),(4,2),(4,26),(4,27),(6,10),(7,8),(8,9),(9,11),(10,12),(11,14),(12,13),(13,5),(14,5),(15,6),(16,20),(16,21),(17,10),(17,20),(18,8),(18,19),(19,9),(19,21),(20,12),(20,22),(21,11),(21,22),(22,13),(22,14),(23,16),(23,19),(24,6),(24,17),(25,16),(25,17),(26,7),(26,18),(26,24),(26,25),(27,7),(27,15),(27,18),(27,23)],28)
=> ? = 60
[4,2,1,1,1]
=> 100101110 => ([(0,4),(0,5),(1,18),(1,32),(2,3),(2,17),(2,31),(3,8),(3,30),(4,1),(4,33),(4,34),(5,2),(5,33),(5,34),(7,14),(8,11),(9,15),(10,16),(11,12),(12,9),(13,10),(14,13),(15,6),(16,6),(17,8),(18,7),(19,21),(19,22),(20,19),(20,27),(21,10),(21,23),(22,9),(22,23),(23,15),(23,16),(24,19),(24,28),(25,20),(25,29),(26,20),(26,24),(27,13),(27,21),(28,12),(28,22),(29,14),(29,27),(30,11),(30,28),(31,24),(31,30),(32,7),(32,29),(33,25),(33,26),(33,31),(33,32),(34,17),(34,18),(34,25),(34,26)],35)
=> ? = 48
[3,3,3]
=> 111000 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> 20
[3,3,2,1]
=> 1101010 => ([(0,2),(0,3),(1,10),(2,14),(2,17),(3,1),(3,14),(3,17),(5,8),(6,5),(7,9),(8,4),(9,4),(10,6),(11,13),(11,16),(12,8),(12,9),(13,7),(13,12),(14,11),(14,15),(15,6),(15,13),(15,16),(16,5),(16,7),(16,12),(17,10),(17,11),(17,15)],18)
=> ? = 60
[3,3,1,1,1]
=> 11001110 => ([(0,4),(0,5),(1,23),(1,25),(2,3),(2,24),(2,26),(3,8),(3,18),(4,1),(4,19),(4,20),(5,2),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,10),(12,9),(13,6),(14,6),(15,13),(15,14),(16,12),(16,21),(17,21),(17,22),(18,11),(18,22),(19,23),(19,24),(20,25),(20,26),(21,9),(21,15),(22,10),(22,15),(23,16),(23,17),(24,17),(24,18),(25,7),(25,16),(26,7),(26,8)],27)
=> ? = 36
Description
The number of maximal chains in a poset.
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