- FindStat

Your data matches 42 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000078
(load all 7 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000078: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => 1
[.,[.,.]]
 => [2,1] => 1
[[.,.],.]
 => [1,2] => 1
[.,[.,[.,.]]]
 => [3,2,1] => 1
[.,[[.,.],.]]
 => [2,3,1] => 1
[[.,.],[.,.]]
 => [1,3,2] => 2
[[.,[.,.]],.]
 => [2,1,3] => 1
[[[.,.],.],.]
 => [1,2,3] => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => 5
[[.,.],[[.,.],.]]
 => [1,3,4,2] => 3
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => 4
[[[.,.],.],[.,.]]
 => [1,2,4,3] => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 5
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 3
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 4
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => 3
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 14
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 9
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 11
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 7
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => 4
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 21
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => 8
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => 14
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => 6
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 8
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => 6
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => 13
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => 5
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => 4

Description

The number of alternating sign matrices whose left key is the permutation. The left key of an alternating sign matrix was defined by Lascoux in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.

Matching statistic: St000455

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 45%●distinct values known / distinct values provided: 7%

Values

[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1 - 2
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => -1 = 1 - 2
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => -1 = 1 - 2
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,4,5} - 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,4,5} - 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,4,5} - 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,4,5} - 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 2
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 1 - 2
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 1 - 2
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 1 - 2
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 1 - 2
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 1 - 2
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 1 - 2
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 1 - 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[[.,[.,.]],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[[.,.],[.,[.,.]]],.]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[[.,.],[[.,.],.]],.]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[[.,[.,.]],[.,.]],.]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[.,[[[[.,.],.],[.,.]],.]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[.,[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[.,[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[.,[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[.,[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2
[[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 2

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001876
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 38%●distinct values known / distinct values provided: 5%

Values

[.,.]
 => ([],1)
 => ([],1)
 => ? = 1 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1} - 1
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1} - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 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
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 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
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? ∊ {3,3,4,5} - 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? ∊ {3,3,4,5} - 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? ∊ {3,3,4,5} - 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? ∊ {3,3,4,5} - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 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
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 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
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 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
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 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
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 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
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 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
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(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),(1,5),(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
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(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),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(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),(1,5),(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
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(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),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(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),(1,5),(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
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(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),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 1

Description

The number of 2-regular simple modules in the incidence algebra of the lattice.

Matching statistic: St000100

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 34%●distinct values known / distinct values provided: 5%

Values

[.,.]
 => ([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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)
 => 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)
 => 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)
 => 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
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(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)
 => ? ∊ {3,3,4,5}
[[.,.],[[.,.],.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[[.,.],.],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,[.,.]]],.]
 => ([(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
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(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
[[[[.,.],.],.],.]
 => ([(0,3),(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),(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
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(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
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[.,[[.,[.,.]],.]]]
 => ([(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
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(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
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,[.,.]]],.]]
 => ([(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
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(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
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,[.,.]],.],.]]
 => ([(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
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(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
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,[.,.]]]],.]
 => ([(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
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(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
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,[.,.]],.]],.]
 => ([(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
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(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
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[[.,.],.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,[.,.]]],.],.]
 => ([(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
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(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
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,[.,.]],.],.],.]
 => ([(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
[[[[[.,.],.],.],.],.]
 => ([(0,4),(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),(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
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(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
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(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
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(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
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(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
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(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
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(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
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(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
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[[.,.],.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,[.,.]],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[[.,.],.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(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
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(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
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(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
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(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
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(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
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(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
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(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
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(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

Description

The number of linear extensions of a poset.

Matching statistic: St000298

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 34%●distinct values known / distinct values provided: 5%

Values

[.,.]
 => ([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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)
 => 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)
 => 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)
 => 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
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(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)
 => ? ∊ {3,3,4,5}
[[.,.],[[.,.],.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[[.,.],.],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,[.,.]]],.]
 => ([(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
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(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
[[[[.,.],.],.],.]
 => ([(0,3),(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),(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
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(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
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[.,[[.,[.,.]],.]]]
 => ([(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
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(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
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,[.,.]]],.]]
 => ([(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
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(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
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,[.,.]],.],.]]
 => ([(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
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(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
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,[.,.]]]],.]
 => ([(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
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(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
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,[.,.]],.]],.]
 => ([(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
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(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
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[[.,.],.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,[.,.]]],.],.]
 => ([(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
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(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
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,[.,.]],.],.],.]
 => ([(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
[[[[[.,.],.],.],.],.]
 => ([(0,4),(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),(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
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(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
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(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
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(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
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(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
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(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
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(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
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(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
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[[.,.],.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,[.,.]],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[[.,.],.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(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
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(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
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(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
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(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
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(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
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(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
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(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
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(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

Description

The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.

Matching statistic: St000307

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 34%●distinct values known / distinct values provided: 5%

Values

[.,.]
 => ([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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)
 => 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)
 => 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)
 => 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
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(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)
 => ? ∊ {3,3,4,5}
[[.,.],[[.,.],.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[[.,.],.],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,[.,.]]],.]
 => ([(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
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(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
[[[[.,.],.],.],.]
 => ([(0,3),(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),(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
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(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
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[.,[[.,[.,.]],.]]]
 => ([(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
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(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
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,[.,.]]],.]]
 => ([(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
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(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
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,[.,.]],.],.]]
 => ([(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
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(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
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,[.,.]]]],.]
 => ([(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
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(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
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,[.,.]],.]],.]
 => ([(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
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(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
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[[.,.],.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,[.,.]]],.],.]
 => ([(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
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(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
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,[.,.]],.],.],.]
 => ([(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
[[[[[.,.],.],.],.],.]
 => ([(0,4),(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),(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
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(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
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(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
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(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
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(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
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(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
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(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
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(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
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[[.,.],.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,[.,.]],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[[.,.],.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(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
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(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
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(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
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(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
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(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
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(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
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(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
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(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

Description

The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.

Matching statistic: St000845

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 34%●distinct values known / distinct values provided: 5%

Values

[.,.]
 => ([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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)
 => 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)
 => 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)
 => 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
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(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)
 => ? ∊ {3,3,4,5}
[[.,.],[[.,.],.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[[.,.],.],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,[.,.]]],.]
 => ([(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
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(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
[[[[.,.],.],.],.]
 => ([(0,3),(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),(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
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(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
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[.,[[.,[.,.]],.]]]
 => ([(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
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(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
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,[.,.]]],.]]
 => ([(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
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(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
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,[.,.]],.],.]]
 => ([(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
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(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
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,[.,.]]]],.]
 => ([(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
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(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
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,[.,.]],.]],.]
 => ([(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
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(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
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[[.,.],.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,[.,.]]],.],.]
 => ([(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
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(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
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,[.,.]],.],.],.]
 => ([(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
[[[[[.,.],.],.],.],.]
 => ([(0,4),(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),(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
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(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
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(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
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(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
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(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
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(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
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(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
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(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
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[[.,.],.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,[.,.]],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[[.,.],.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(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
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(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
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(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
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(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
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(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
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(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
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(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
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(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

Description

The maximal number of elements covered by an element in a poset.

Matching statistic: St000846

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 34%●distinct values known / distinct values provided: 5%

Values

[.,.]
 => ([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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)
 => 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)
 => 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)
 => 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
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(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)
 => ? ∊ {3,3,4,5}
[[.,.],[[.,.],.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[[.,.],.],[.,.]]
 => ([(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)
 => ? ∊ {3,3,4,5}
[[.,[.,[.,.]]],.]
 => ([(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
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(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
[[[[.,.],.],.],.]
 => ([(0,3),(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),(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
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(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
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[.,[[.,[.,.]],.]]]
 => ([(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
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(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
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[.,[.,[.,.]]],.]]
 => ([(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
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(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
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[[[.,[.,.]],.],.]]
 => ([(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
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(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
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,[.,.]]]],.]
 => ([(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
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(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
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,[.,.]],.]],.]
 => ([(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
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(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
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[[.,.],.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],[.,.]],.]
 => ([(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,[.,.]]],.],.]
 => ([(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
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(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
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,[.,.]],.],.],.]
 => ([(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
[[[[[.,.],.],.],.],.]
 => ([(0,4),(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),(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
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(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
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(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
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(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
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(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
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(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
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(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
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(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
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,.],[[[.,.],.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],.],[[.,.],.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[.,[.,.]],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[[[.,.],.],.],[.,.]]]
 => ([(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(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
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(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
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(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
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(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
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(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
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(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
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(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
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(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

Description

The maximal number of elements covering an element of a poset.

Matching statistic: St000632

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 34%●distinct values known / distinct values provided: 5%

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
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 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
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 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
[.,[.,[[.,.],.]]]
 => ([(0,3),(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
[.,[[.,.],[.,.]]]
 => ([(0,3),(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
[.,[[[.,.],.],.]]
 => ([(0,3),(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
[[.,.],[.,[.,.]]]
 => ([(0,3),(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)
 => ? ∊ {3,3,4,5} - 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)
 => ? ∊ {3,3,4,5} - 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)
 => ? ∊ {3,3,4,5} - 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)
 => ? ∊ {3,3,4,5} - 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
[[.,[[.,.],.]],.]
 => ([(0,3),(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
[[[.,.],[.,.]],.]
 => ([(0,3),(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
[[[[.,.],.],.],.]
 => ([(0,3),(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),(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
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(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
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(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
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(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
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(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
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(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
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(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
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(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
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(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)
 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21} - 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
[[[[[.,.],.],.],.],.]
 => ([(0,4),(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),(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
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(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
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(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
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(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
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(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
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135} - 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
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(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
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(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
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(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
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(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
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(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
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(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
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(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

Description

The jump number of the poset. A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

Matching statistic: St000772
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 33%●distinct values known / distinct values provided: 10%

Values

[.,.]
 => [1] => [1] => ([],1)
 => 1
[.,[.,.]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[.,.],.]
 => [1,2] => [1,2] => ([],2)
 => ? = 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1,2}
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,1,2}
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {1,1,2}
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {1,1,1,1,1,3,3,4,5}
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,5,5,6,6,7,8,8,9,11,13,14,14,21}
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => [6,1,4,3,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [6,1,5,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [5,2,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => [4,6,1,3,5,2] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => [4,5,2,6,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => [5,2,4,6,1,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => [3,6,1,5,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [3,5,4,6,2,1] => [5,2,6,1,3,4] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => [6,1,3,5,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [2,6,5,4,3,1] => [4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [2,5,6,4,3,1] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [2,4,6,5,3,1] => [6,1,2,4,5,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[.,[[.,.],[[.,[.,.]],.]]]
 => [2,5,4,6,3,1] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[.,[[.,.],[[[.,.],.],.]]]
 => [2,4,5,6,3,1] => [5,3,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [3,2,6,5,4,1] => [2,5,4,6,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [3,2,5,6,4,1] => [2,6,1,3,5,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [2,3,6,5,4,1] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[.,[[[.,.],.],[[.,.],.]]]
 => [2,3,5,6,4,1] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[.,[[.,[.,[.,.]]],[.,.]]]
 => [4,3,2,6,5,1] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[.,[[.,[[.,.],.]],[.,.]]]
 => [3,4,2,6,5,1] => [5,6,1,3,2,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[.,[[[.,.],[.,.]],[.,.]]]
 => [2,4,3,6,5,1] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [3,2,4,6,5,1] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => [1,4,5,3,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[[.,.],[.,[.,[[.,.],.]]]]
 => [1,5,6,4,3,2] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[[.,.],[.,[[.,.],[.,.]]]]
 => [1,4,6,5,3,2] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[[.,.],[.,[[.,[.,.]],.]]]
 => [1,5,4,6,3,2] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[[.,.],[.,[[[.,.],.],.]]]
 => [1,4,5,6,3,2] => [1,5,3,6,2,4] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,3,6,5,4,2] => [1,5,4,6,2,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[[.,.],[[.,.],[[.,.],.]]]
 => [1,3,5,6,4,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[[.,.],[[.,[.,.]],[.,.]]]
 => [1,4,3,6,5,2] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}
[[.,.],[[[.,.],.],[.,.]]]
 => [1,3,4,6,5,2] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,14,14,15,15,16,19,19,20,21,21,23,25,26,26,26,28,29,30,32,35,37,38,40,41,42,42,51,53,56,58,60,67,82,84,121,135}

Description

The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].

The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. 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. St001330The hat guessing number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001846The number of elements which do not have a complement in the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. 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. 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. 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. 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. St001875The number of simple modules with projective dimension at most 1. 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. St001438The number of missing boxes of a skew partition.