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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000063
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
[6]
=> 7
[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
[1,1,1,1,1,1]
=> 7
[7]
=> 8
[6,1]
=> 14
[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
[2,1,1,1,1,1]
=> 14
[1,1,1,1,1,1,1]
=> 8
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 2 compositions to match this statistic)
(load all 2 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000085: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[[[]]]]]],[]]
=> 7
[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
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[],[[[[[[]]]]]]]
=> 7
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[]]]]]]],[]]
=> 8
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[[[],[]]]]],[]]
=> 14
[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
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[],[[[[[],[]]]]]]
=> 14
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[],[[[[[[[]]]]]]]]
=> 8
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: St000110
(load all 5 compositions to match this statistic)
(load all 5 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ 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]
=> [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
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 7
[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
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 7
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => 8
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 14
[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
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => 14
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 8
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
(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: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 87%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 87%
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
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? ∊ {7,7}
[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
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? ∊ {7,7}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? ∊ {8,8,14,14}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ? ∊ {8,8,14,14}
[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
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ? ∊ {8,8,14,14}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? ∊ {8,8,14,14}
Description
The number of linear extensions of a poset.
Matching statistic: St001855
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 47%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 47%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 2
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 4
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 6
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 4
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? ∊ {5,5}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,2,1,3] => 8
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 6
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 8
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? ∊ {6,6,10,10}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? ∊ {6,6,10,10}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,1,2] => 12
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 12
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => 12
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? ∊ {6,6,10,10}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? ∊ {6,6,10,10}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [6,2,1,3,4,5] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,2,3,1,4] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 24
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,5,1] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [7,2,1,3,4,5,6] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,2,4,5] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [6,2,3,1,4,5] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,3,2,1,4] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,5,2,1,3] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [4,2,3,5,6,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,4,2,5,6,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,8,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
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: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Mp00262: Binary words —poset of factors⟶ Posets
St000071: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
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)
=> ? ∊ {8,8}
[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)
=> ? ∊ {8,8}
[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)
=> ? ∊ {10,10,12,12,12}
[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)
=> ? ∊ {10,10,12,12,12}
[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)
=> ? ∊ {10,10,12,12,12}
[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)
=> ? ∊ {10,10,12,12,12}
[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)
=> ? ∊ {10,10,12,12,12}
[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
[6]
=> 1000000 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[1,1,1,1,1,1]
=> 1111110 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? ∊ {7,7,12,12,15,15,15,15,24}
[7]
=> 10000000 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[6,1]
=> 10000010 => ([(0,5),(0,6),(1,4),(1,11),(1,20),(2,16),(2,22),(3,2),(3,17),(3,23),(4,3),(4,15),(4,21),(5,18),(5,19),(6,1),(6,18),(6,19),(8,9),(9,13),(10,8),(11,15),(12,10),(13,7),(14,7),(15,17),(16,14),(17,16),(18,11),(18,12),(19,12),(19,20),(20,10),(20,21),(21,8),(21,23),(22,13),(22,14),(23,9),(23,22)],24)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,1,1,1,1,1]
=> 10111110 => ([(0,5),(0,6),(1,4),(1,11),(1,20),(2,16),(2,22),(3,2),(3,17),(3,23),(4,3),(4,15),(4,21),(5,18),(5,19),(6,1),(6,18),(6,19),(8,9),(9,13),(10,8),(11,15),(12,10),(13,7),(14,7),(15,17),(16,14),(17,16),(18,11),(18,12),(19,12),(19,20),(20,10),(20,21),(21,8),(21,23),(22,13),(22,14),(23,9),(23,22)],24)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[1,1,1,1,1,1,1]
=> 11111110 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
Description
The number of maximal chains in a poset.
Matching statistic: St000718
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Values
[1]
=> [1,0]
=> [[]]
=> ([(0,1)],2)
=> 2
[2]
=> [1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 3
[1,1]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 3
[3]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 4
[2,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 6
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 4
[4]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {6,8,8}
[2,2]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,8,8}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {6,8,8}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {10,10,12,12,12}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {10,10,12,12,12}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {10,10,12,12,12}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {10,10,12,12,12}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {10,10,12,12,12}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[],[[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {10,10,12,12,15,15,15,15,24}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[],[],[[[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[],[[[]],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[],[[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St001645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 47%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 47%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1 = 2 - 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 3 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {4,6} - 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {4,6} - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {5,6,8,8} - 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {5,6,8,8} - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {5,6,8,8} - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,6,8,8} - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {6,10,10,12,12,12} - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {6,10,10,12,12,12} - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {6,10,10,12,12,12} - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,10,10,12,12,12} - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,10,10,12,12,12} - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,10,10,12,12,12} - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 40%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 40%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 2 - 2
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 3 - 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 3 - 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {4,4,6} - 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {4,4,6} - 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {4,4,6} - 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,6,8,8} - 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {5,6,8,8} - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,6,8,8} - 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,6,8,8} - 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 6 - 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {6,10,10,12,12,12} - 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {6,10,10,12,12,12} - 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {6,10,10,12,12,12} - 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {6,10,10,12,12,12} - 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {6,10,10,12,12,12} - 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {6,10,10,12,12,12} - 2
[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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 7 - 2
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {7,10,10,12,12,15,15,15,15,24} - 2
[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]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 8 - 2
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {8,14,14,18,18,18,18,20,20,20,20,20,30,30} - 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000909
(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
St000909: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 27%
Mp00262: Binary words —poset of factors⟶ Posets
St000909: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 27%
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)
=> ? ∊ {5,5,8,8}
[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)
=> ? ∊ {5,5,8,8}
[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)
=> ? ∊ {5,5,8,8}
[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)
=> ? ∊ {5,5,8,8}
[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)
=> ? ∊ {6,6,10,10,12,12,12}
[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)
=> ? ∊ {6,6,10,10,12,12,12}
[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)
=> ? ∊ {6,6,10,10,12,12,12}
[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)
=> ? ∊ {6,6,10,10,12,12,12}
[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)
=> ? ∊ {6,6,10,10,12,12,12}
[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)
=> ? ∊ {6,6,10,10,12,12,12}
[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)
=> ? ∊ {6,6,10,10,12,12,12}
[6]
=> 1000000 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[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)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[1,1,1,1,1,1]
=> 1111110 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? ∊ {7,7,10,10,12,12,15,15,15,15,24}
[7]
=> 10000000 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[6,1]
=> 10000010 => ([(0,5),(0,6),(1,4),(1,11),(1,20),(2,16),(2,22),(3,2),(3,17),(3,23),(4,3),(4,15),(4,21),(5,18),(5,19),(6,1),(6,18),(6,19),(8,9),(9,13),(10,8),(11,15),(12,10),(13,7),(14,7),(15,17),(16,14),(17,16),(18,11),(18,12),(19,12),(19,20),(20,10),(20,21),(21,8),(21,23),(22,13),(22,14),(23,9),(23,22)],24)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,1,1,1,1,1]
=> 10111110 => ([(0,5),(0,6),(1,4),(1,11),(1,20),(2,16),(2,22),(3,2),(3,17),(3,23),(4,3),(4,15),(4,21),(5,18),(5,19),(6,1),(6,18),(6,19),(8,9),(9,13),(10,8),(11,15),(12,10),(13,7),(14,7),(15,17),(16,14),(17,16),(18,11),(18,12),(19,12),(19,20),(20,10),(20,21),(21,8),(21,23),(22,13),(22,14),(23,9),(23,22)],24)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[1,1,1,1,1,1,1]
=> 11111110 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
=> ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
Description
The number of maximal chains of maximal size in a poset.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000080The rank of the poset. St000570The Edelman-Greene number of a permutation. St000572The dimension exponent of a set partition. St001298The number of repeated entries in the Lehmer code of a permutation. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001760The number of prefix or suffix reversals needed to sort a permutation. St000222The number of alignments in the permutation. St000516The number of stretching pairs of a permutation. St000519The largest length of a factor maximising the subword complexity. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000906The length of the shortest maximal chain in a poset. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001375The pancake length of a permutation. St001535The number of cyclic alignments of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001841The number of inversions of a set partition. St001911A descent variant minus the number of inversions. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001565The number of arithmetic progressions of length 2 in a permutation. St001782The order of rowmotion on the set of order ideals of a poset.
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