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Your data matches 41 different statistics following compositions of up to 3 maps.
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Matching statistic: St000063
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Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> []
=> 1
[[]]
=> [1,0]
=> []
=> 1
[[],[]]
=> [1,0,1,0]
=> [1]
=> 2
[[[]]]
=> [1,1,0,0]
=> []
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 6
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 3
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2]
=> 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 8
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 4
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 6
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 8
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 6
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 10
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 5
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 10
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 10
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 5
[[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 8
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 4
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 6
[[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 8
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 4
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 6
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 3
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 3
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 2
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[[],[[[[[]]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 6
[[[[[[]]]]],[]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 6
[[[],[[[],[]]]]]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 10
[[[],[[[[]]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 5
[[[[]],[[[]]]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 10
[[[[[]]],[[]]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 10
[[[[[],[]]],[]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 10
[[[[[[]]]],[]]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 5
[[[[],[[],[]]]]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 8
[[[[],[[[]]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 4
[[[[[]],[[]]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 6
[[[[[],[]],[]]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 8
[[[[[[]]],[]]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 4
[[[[[],[],[]]]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 6
[[[[[],[[]]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 3
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: St000228
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Mp00047: Ordered trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> [1]
=> 1
[[]]
=> ([(0,1)],2)
=> [1]
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 4
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 10
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 5
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> 10
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> 10
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 5
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 8
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 4
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 6
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 8
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 4
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [3,3]
=> 6
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [6]
=> 6
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [6]
=> 6
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [5,5]
=> 10
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> 5
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [5,5]
=> 10
[[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [5,5]
=> 10
[[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [5,5]
=> 10
[[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> 5
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [8]
=> 8
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> [4]
=> 4
[[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> [4,2]
=> 6
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [8]
=> 8
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> [4]
=> 4
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [3,3]
=> 6
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> [3]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000085
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(load all 3 compositions to match this statistic)
St000085: Ordered trees ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[]
=> ? = 1
[[]]
=> 1
[[],[]]
=> 2
[[[]]]
=> 1
[[],[],[]]
=> 6
[[],[[]]]
=> 3
[[[]],[]]
=> 3
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[[],[]]]
=> 8
[[],[[[]]]]
=> 4
[[[]],[[]]]
=> 6
[[[],[]],[]]
=> 8
[[[[]]],[]]
=> 4
[[[],[],[]]]
=> 6
[[[],[[]]]]
=> 3
[[[[]],[]]]
=> 3
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[[[],[]]]]
=> 10
[[],[[[[]]]]]
=> 5
[[[]],[[[]]]]
=> 10
[[[[]]],[[]]]
=> 10
[[[[],[]]],[]]
=> 10
[[[[[]]]],[]]
=> 5
[[[],[[],[]]]]
=> 8
[[[],[[[]]]]]
=> 4
[[[[]],[[]]]]
=> 6
[[[[],[]],[]]]
=> 8
[[[[[]]],[]]]
=> 4
[[[[],[],[]]]]
=> 6
[[[[],[[]]]]]
=> 3
[[[[[]],[]]]]
=> 3
[[[[[],[]]]]]
=> 2
[[[[[[]]]]]]
=> 1
[[],[[[[[]]]]]]
=> 6
[[[[[[]]]]],[]]
=> 6
[[[],[[[],[]]]]]
=> 10
[[[],[[[[]]]]]]
=> 5
[[[[]],[[[]]]]]
=> 10
[[[[[]]],[[]]]]
=> 10
[[[[[],[]]],[]]]
=> 10
[[[[[[]]]],[]]]
=> 5
[[[[],[[],[]]]]]
=> 8
[[[[],[[[]]]]]]
=> 4
[[[[[]],[[]]]]]
=> 6
[[[[[],[]],[]]]]
=> 8
[[[[[[]]],[]]]]
=> 4
[[[[[],[],[]]]]]
=> 6
[[[[[],[[]]]]]]
=> 3
[[[[[[]],[]]]]]
=> 3
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: St000100
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
St000100: Posets ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> ? = 1
[[]]
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 10
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 10
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 10
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 8
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 4
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 6
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 8
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 4
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 6
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 6
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 6
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> 10
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> 5
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> 10
[[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> 10
[[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> 10
[[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> 5
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> 8
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> 4
[[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> 6
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> 8
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> 4
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> 6
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> 3
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> 3
Description
The number of linear extensions of a poset.
Matching statistic: St000110
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[]
=> .
=> ? => ? = 1
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 6
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 3
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => 3
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 8
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 4
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 6
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 8
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 4
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 6
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 10
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 5
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 10
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 10
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 10
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 5
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 8
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 4
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 6
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 8
[[[[[]]],[]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 4
[[[[],[],[]]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 6
[[[[],[[]]]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 3
[[[[[]],[]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 3
[[[[[],[]]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 2
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 1
[[],[[[[[]]]]]]
=> [.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => 6
[[[[[[]]]]],[]]
=> [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 6
[[[],[[[],[]]]]]
=> [[.,[[[.,[.,.]],.],.]],.]
=> [3,2,4,5,1,6] => 10
[[[],[[[[]]]]]]
=> [[.,[[[[.,.],.],.],.]],.]
=> [2,3,4,5,1,6] => 5
[[[[]],[[[]]]]]
=> [[[.,.],[[[.,.],.],.]],.]
=> [3,4,5,1,2,6] => 10
[[[[[]]],[[]]]]
=> [[[[.,.],.],[[.,.],.]],.]
=> [4,5,1,2,3,6] => 10
[[[[[],[]]],[]]]
=> [[[[.,[.,.]],.],[.,.]],.]
=> [5,2,1,3,4,6] => 10
[[[[[[]]]],[]]]
=> [[[[[.,.],.],.],[.,.]],.]
=> [5,1,2,3,4,6] => 5
[[[[],[[],[]]]]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> [3,2,4,1,5,6] => 8
[[[[],[[[]]]]]]
=> [[[.,[[[.,.],.],.]],.],.]
=> [2,3,4,1,5,6] => 4
[[[[[]],[[]]]]]
=> [[[[.,.],[[.,.],.]],.],.]
=> [3,4,1,2,5,6] => 6
[[[[[],[]],[]]]]
=> [[[[.,[.,.]],[.,.]],.],.]
=> [4,2,1,3,5,6] => 8
[[[[[[]]],[]]]]
=> [[[[[.,.],.],[.,.]],.],.]
=> [4,1,2,3,5,6] => 4
[[[[[],[],[]]]]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> [3,2,1,4,5,6] => 6
[[[[[],[[]]]]]]
=> [[[[.,[[.,.],.]],.],.],.]
=> [2,3,1,4,5,6] => 3
[[[[[[]],[]]]]]
=> [[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => 3
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: St000071
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 10
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> 5
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> 10
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> 10
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> 5
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 8
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> 4
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> 6
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 8
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> 4
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 6
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 6
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 6
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ? = 10
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ? = 5
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ? = 10
[[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ? = 10
[[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ? = 10
[[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ? = 5
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ? = 8
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ? = 4
[[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,6),(0,7),(2,10),(3,9),(4,1),(5,4),(6,3),(6,8),(7,2),(7,8),(8,9),(8,10),(9,11),(10,11),(11,5)],12)
=> ([(0,6),(0,7),(2,10),(3,9),(4,1),(5,4),(6,3),(6,8),(7,2),(7,8),(8,9),(8,10),(9,11),(10,11),(11,5)],12)
=> ? = 6
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ? = 8
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ? = 4
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,6),(6,1),(7,5),(8,11),(9,11),(10,11),(11,7)],12)
=> ([(0,2),(0,3),(0,4),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,6),(6,1),(7,5),(8,11),(9,11),(10,11),(11,7)],12)
=> ? = 6
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> 3
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> 3
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> 2
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
Description
The number of maximal chains in a poset.
Matching statistic: St001855
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 75%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 75%
Values
[]
=> .
=> ? => ? => ? = 1
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [2,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 6
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 3
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 3
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 8
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 4
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 6
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => 8
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => 4
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 6
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 10
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 5
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,5,1,2] => ? = 10
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 10
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 10
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 5
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 8
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 4
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 6
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [4,2,1,3,5] => ? = 8
[[[[[]]],[]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 4
[[[[],[],[]]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 6
[[[[],[[]]]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,3,1,4,5] => ? = 3
[[[[[]],[]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 3
[[[[[],[]]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[[],[[[[[]]]]]]
=> [.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 6
[[[[[[]]]]],[]]
=> [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 6
[[[],[[[],[]]]]]
=> [[.,[[[.,[.,.]],.],.]],.]
=> [3,2,4,5,1,6] => [3,2,4,5,1,6] => ? = 10
[[[],[[[[]]]]]]
=> [[.,[[[[.,.],.],.],.]],.]
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => ? = 5
[[[[]],[[[]]]]]
=> [[[.,.],[[[.,.],.],.]],.]
=> [3,4,5,1,2,6] => [3,4,5,1,2,6] => ? = 10
[[[[[]]],[[]]]]
=> [[[[.,.],.],[[.,.],.]],.]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 10
[[[[[],[]]],[]]]
=> [[[[.,[.,.]],.],[.,.]],.]
=> [5,2,1,3,4,6] => [5,2,1,3,4,6] => ? = 10
[[[[[[]]]],[]]]
=> [[[[[.,.],.],.],[.,.]],.]
=> [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 5
[[[[],[[],[]]]]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> [3,2,4,1,5,6] => [3,2,4,1,5,6] => ? = 8
[[[[],[[[]]]]]]
=> [[[.,[[[.,.],.],.]],.],.]
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => ? = 4
[[[[[]],[[]]]]]
=> [[[[.,.],[[.,.],.]],.],.]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 6
[[[[[],[]],[]]]]
=> [[[[.,[.,.]],[.,.]],.],.]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => ? = 8
[[[[[[]]],[]]]]
=> [[[[[.,.],.],[.,.]],.],.]
=> [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 4
[[[[[],[],[]]]]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? = 6
[[[[[],[[]]]]]]
=> [[[[.,[[.,.],.]],.],.],.]
=> [2,3,1,4,5,6] => [2,3,1,4,5,6] => ? = 3
[[[[[[]],[]]]]]
=> [[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 3
[[[[[[],[]]]]]]
=> [[[[[.,[.,.]],.],.],.],.]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 2
[[[[[[[]]]]]]]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
Matching statistic: St000909
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 10
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 5
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 10
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 10
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 5
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 8
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 6
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 8
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 6
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 6
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 6
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ? = 10
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ? = 5
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ? = 10
[[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ? = 10
[[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ? = 10
[[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ? = 5
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ? = 8
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ? = 4
[[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,6),(0,7),(2,10),(3,9),(4,1),(5,4),(6,3),(6,8),(7,2),(7,8),(8,9),(8,10),(9,11),(10,11),(11,5)],12)
=> ([(0,6),(0,7),(2,10),(3,9),(4,1),(5,4),(6,3),(6,8),(7,2),(7,8),(8,9),(8,10),(9,11),(10,11),(11,5)],12)
=> ? = 6
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ? = 8
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ? = 4
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,6),(6,1),(7,5),(8,11),(9,11),(10,11),(11,7)],12)
=> ([(0,2),(0,3),(0,4),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,6),(6,1),(7,5),(8,11),(9,11),(10,11),(11,7)],12)
=> ? = 6
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 3
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 3
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ? = 2
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St001633
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 6 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 3 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 3 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 2 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 8 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 6 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 8 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 6 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 3 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 3 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 2 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 10 - 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 5 - 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 10 - 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 10 - 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 10 - 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 5 - 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 8 - 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 - 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 6 - 1
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 8 - 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 - 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 6 - 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 3 - 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 3 - 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2 - 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 6 - 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 6 - 1
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ? = 10 - 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ? = 5 - 1
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ? = 10 - 1
[[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ? = 10 - 1
[[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
=> ? = 10 - 1
[[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ([(0,3),(0,7),(2,11),(3,8),(4,1),(5,6),(5,10),(6,2),(6,9),(7,5),(7,8),(8,10),(9,11),(10,9),(11,4)],12)
=> ? = 5 - 1
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ? = 8 - 1
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ? = 4 - 1
[[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,6),(0,7),(2,10),(3,9),(4,1),(5,4),(6,3),(6,8),(7,2),(7,8),(8,9),(8,10),(9,11),(10,11),(11,5)],12)
=> ([(0,6),(0,7),(2,10),(3,9),(4,1),(5,4),(6,3),(6,8),(7,2),(7,8),(8,9),(8,10),(9,11),(10,11),(11,5)],12)
=> ? = 6 - 1
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
=> ? = 8 - 1
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ([(0,3),(0,7),(2,10),(3,8),(4,5),(5,1),(6,2),(6,9),(7,6),(7,8),(8,9),(9,10),(10,4)],11)
=> ? = 4 - 1
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,6),(6,1),(7,5),(8,11),(9,11),(10,11),(11,7)],12)
=> ([(0,2),(0,3),(0,4),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,6),(6,1),(7,5),(8,11),(9,11),(10,11),(11,7)],12)
=> ? = 6 - 1
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 3 - 1
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 3 - 1
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ? = 2 - 1
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> ? = 1 - 1
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? = 6 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ? = 3 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ? = 3 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 8 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ? = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ? = 6 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 8 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ? = 4 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ? = 6 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 3 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 3 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 2 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ? = 10 - 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 5 - 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ? = 10 - 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ? = 10 - 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ? = 10 - 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 5 - 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ? = 8 - 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 4 - 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ? = 6 - 1
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ? = 8 - 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 4 - 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 6 - 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 3 - 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 3 - 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 2 - 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,2),(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,4)],8)
=> ? = 6 - 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,2),(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,4)],8)
=> ? = 6 - 1
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,7),(3,6),(4,6),(5,7),(6,2),(7,1)],8)
=> ? = 10 - 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 5 - 1
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 10 - 1
[[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 10 - 1
[[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,7),(3,6),(4,6),(5,7),(6,2),(7,1)],8)
=> ? = 10 - 1
[[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 5 - 1
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,7),(3,6),(4,6),(5,1),(6,7),(7,5)],8)
=> ? = 8 - 1
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 4 - 1
[[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 6 - 1
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,7),(3,6),(4,6),(5,1),(6,7),(7,5)],8)
=> ? = 8 - 1
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 4 - 1
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,7),(3,7),(4,7),(5,1),(6,5),(7,6)],8)
=> ? = 6 - 1
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 3 - 1
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 3 - 1
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 1 = 2 - 1
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001877Number of indecomposable injective modules with projective dimension 2. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000632The jump number of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001330The hat guessing number of a graph. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001964The interval resolution global dimension of a poset.
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