Your data matches 48 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00026: Dyck paths to ordered treeOrdered 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 $v$.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
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.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00065: Permutations permutation posetPosets
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 pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00170: Permutations to signed permutationSigned 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.
Mp00095: Integer partitions to binary wordBinary words
Mp00262: Binary words poset of factorsPosets
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 polyominoDyck paths
Mp00026: Dyck paths to ordered treeOrdered trees
Mp00046: Ordered trees to graphGraphs
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]
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
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.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
Mp00103: Dyck paths peeling mapDyck 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.
Mp00095: Integer partitions to binary wordBinary words
Mp00262: Binary words poset of factorsPosets
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.