Your data matches 56 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001554
(load all 2 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => 1 = 2 - 1
[1,2] => [.,[.,.]]
 => 2 = 3 - 1
[2,1] => [[.,.],.]
 => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
 => 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
 => 3 = 4 - 1
[2,1,3] => [[.,.],[.,.]]
 => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
 => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
 => 3 = 4 - 1
[3,2,1] => [[[.,.],.],.]
 => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => 4 = 5 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => 4 = 5 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => 4 = 5 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => 3 = 4 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => 3 = 4 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => 3 = 4 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => 4 = 5 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => 4 = 5 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => 4 = 5 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 6 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 5 = 6 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => 5 = 6 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => 5 = 6 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => 4 = 5 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => 4 = 5 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => 4 = 5 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => 4 = 5 - 1
Description
The number of distinct nonempty subtrees of a binary tree.
Matching statistic: St000528
(load all 3 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => 1 = 2 - 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
Description
The height of a poset. This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000912
(load all 2 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => 1 = 2 - 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
Description
The number of maximal antichains in a poset.
Matching statistic: St001343
(load all 3 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => 1 = 2 - 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4 = 5 - 1
Description
The dimension of the reduced incidence algebra of a poset. The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets. Thus, this statistic returns the number of non-isomorphic intervals of the poset.
Matching statistic: St000380
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => [1]
 => 2
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => [2]
 => 3
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => [2]
 => 3
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 4
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 4
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 6
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 6
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 6
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 6
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 5
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St001619
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001619: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => ([],1)
 => 2
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 3
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 3
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 3
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 3
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
Description
The number of non-isomorphic sublattices of a lattice.
Matching statistic: St001666
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001666: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => ([],1)
 => 2
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 3
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 3
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 3
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 3
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5
Description
The number of non-isomorphic subposets of a lattice which are lattices.
Matching statistic: St001720
(load all 2 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => ([(0,1)],2)
 => 2
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(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)
 => 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(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)
 => 5
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(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)
 => 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(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)
 => 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(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)
 => 5
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(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)
 => 5
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(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)
 => 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(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)
 => 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => 5
Description
The minimal length of a chain of small intervals in a lattice. An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St000093
(load all 2 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 2 = 3 - 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3 = 4 - 1
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3 = 4 - 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4 = 5 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4 = 5 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4 = 5 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4 = 5 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4 = 5 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4 = 5 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5 = 6 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5 = 6 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5 = 6 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5 = 6 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000147
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => ([],1)
 => [1]
 => 1 = 2 - 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => [2]
 => 2 = 3 - 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => [2]
 => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3 = 4 - 1
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3 = 4 - 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4 = 5 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4 = 5 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4 = 5 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4 = 5 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4 = 5 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4 = 5 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5 = 6 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5 = 6 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5 = 6 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5 = 6 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 4 = 5 - 1
Description
The largest part of an integer partition.
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000184The size of the centralizer of any permutation of given cycle type. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000636The hull number of a graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001616The number of neutral elements in a lattice. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001820The size of the image of the pop stack sorting operator. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001176The size of a partition minus its first part. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001651The Frankl number of a lattice. St001875The number of simple modules with projective dimension at most 1. St000672The number of minimal elements in Bruhat order not less than the permutation. St001717The largest size of an interval in a poset. St000050The depth or height of a binary tree. St000863The length of the first row of the shifted shape of a permutation. St000553The number of blocks of a graph. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000080The rank of the poset. St000013The height of a Dyck path. St001322The size of a minimal independent dominating set in a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001668The number of points of the poset minus the width of the poset. St001626The number of maximal proper sublattices of a lattice. St001060The distinguishing index of a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation.