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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000032
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> 1
[.,[.,.]]
=> [1,1,0,0]
=> 2
[[.,.],.]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 5
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 3
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 14
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 9
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 7
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 7
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 4
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 5
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 4
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 5
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 42
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 28
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 23
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 23
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 14
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 19
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 12
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 16
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 9
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 19
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 12
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 9
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 9
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 14
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 9
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 7
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 7
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 10
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 6
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 10
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 6
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
Description
The number of elements smaller than the given Dyck path in the Tamari Order.
Matching statistic: St000082
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(load all 3 compositions to match this statistic)
St000082: Binary trees ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 44%
Values
[.,.]
=> ? = 1
[.,[.,.]]
=> 2
[[.,.],.]
=> 1
[.,[.,[.,.]]]
=> 5
[.,[[.,.],.]]
=> 3
[[.,.],[.,.]]
=> 2
[[.,[.,.]],.]
=> 2
[[[.,.],.],.]
=> 1
[.,[.,[.,[.,.]]]]
=> 14
[.,[.,[[.,.],.]]]
=> 9
[.,[[.,.],[.,.]]]
=> 7
[.,[[.,[.,.]],.]]
=> 7
[.,[[[.,.],.],.]]
=> 4
[[.,.],[.,[.,.]]]
=> 5
[[.,.],[[.,.],.]]
=> 3
[[.,[.,.]],[.,.]]
=> 4
[[[.,.],.],[.,.]]
=> 2
[[.,[.,[.,.]]],.]
=> 5
[[.,[[.,.],.]],.]
=> 3
[[[.,.],[.,.]],.]
=> 2
[[[.,[.,.]],.],.]
=> 2
[[[[.,.],.],.],.]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> 42
[.,[.,[.,[[.,.],.]]]]
=> 28
[.,[.,[[.,.],[.,.]]]]
=> 23
[.,[.,[[.,[.,.]],.]]]
=> 23
[.,[.,[[[.,.],.],.]]]
=> 14
[.,[[.,.],[.,[.,.]]]]
=> 19
[.,[[.,.],[[.,.],.]]]
=> 12
[.,[[.,[.,.]],[.,.]]]
=> 16
[.,[[[.,.],.],[.,.]]]
=> 9
[.,[[.,[.,[.,.]]],.]]
=> 19
[.,[[.,[[.,.],.]],.]]
=> 12
[.,[[[.,.],[.,.]],.]]
=> 9
[.,[[[.,[.,.]],.],.]]
=> 9
[.,[[[[.,.],.],.],.]]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> 14
[[.,.],[.,[[.,.],.]]]
=> 9
[[.,.],[[.,.],[.,.]]]
=> 7
[[.,.],[[.,[.,.]],.]]
=> 7
[[.,.],[[[.,.],.],.]]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> 10
[[.,[.,.]],[[.,.],.]]
=> 6
[[[.,.],.],[.,[.,.]]]
=> 5
[[[.,.],.],[[.,.],.]]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> 10
[[.,[[.,.],.]],[.,.]]
=> 6
[[[.,.],[.,.]],[.,.]]
=> 4
[[[.,[.,.]],.],[.,.]]
=> 4
[[[[.,.],.],.],[.,.]]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> 14
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[.,[[[[[.,.],.],.],.],.]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,27,27,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,28,29,29,29,29,30,30,30,30,30,30,30,30,32,32,33,33,33,33,33,33,34,34,34,34,34,34,35,35,35,35,37,37,37,37,37,37,37,37,37,37,38,38,38,38,40,40,42,42,42,42,42,43,43,43,43,43,43,43,43,43,43,43,43,43,43,44,45,45,45,46,46,46,46,46,46,46,47,47,47,47,48,48,53,53,53,53,53,53,55,55,55,55,56,56,56,56,56,56,57,57,61,61,61,62,62,66,66,66,66,67,67,67,67,70,70,70,70,70,70,70,72,72,73,73,75,76,76,76,76,81,81,81,83,83,84,84,85,85,85,85,90,90,95,95,95,99,99,99,99,104,104,113,113,113,113,115,118,118,123,123,123,126,126,132,132,136,136,151,151,161,161,165,174,174,199,202,202,227,227,255,255,297,429}
Description
The number of elements smaller than a binary tree in Tamari order.
Matching statistic: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 10%●distinct values known / distinct values provided: 9%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 10%●distinct values known / distinct values provided: 9%
Values
[.,.]
=> [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> ? = 5 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {4,5,5,7,7,9,14} - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {4,5,5,7,7,9,14} - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {4,5,5,7,7,9,14} - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {4,5,5,7,7,9,14} - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {4,5,5,7,7,9,14} - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {4,5,5,7,7,9,14} - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,5,5,7,7,9,14} - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [[[[[.,.],.],.],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [[[[[.,.],.],[.,.]],.],.]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [[[[[.,.],.],.],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [6,3,4,5,2,1] => [[[[.,.],.],[.,[.,.]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [5,3,4,6,2,1] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[[[[.,.],.],.],.]]]
=> [3,4,5,6,2,1] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => [[[[[.,.],[.,.]],.],.],.]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [5,6,4,2,3,1] => [[[[.,.],[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [5,4,6,2,3,1] => [[[[.,.],[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[[.,.],[[[.,.],.],.]]]
=> [4,5,6,2,3,1] => [[[.,.],[.,.]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? ∊ {4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [6,5,3,2,4,1] => [[[[[.,.],.],[.,.]],.],.]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [6,4,3,2,5,1] => [[[[[.,.],.],.],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> [5,4,3,2,6,1] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> [6,5,4,2,1,3] => [[[[[.,.],[.,.]],.],.],.]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> [6,5,3,2,1,4] => [[[[[.,.],.],[.,.]],.],.]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[[.,[.,[.,[.,.]]]],[.,.]]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 15%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 15%
Values
[.,.]
=> [1] => [1] => ([],1)
=> ? = 1
[.,[.,.]]
=> [2,1] => [2,1] => ([],2)
=> ? ∊ {1,2}
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,2}
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,2,3,5}
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,2,3,5}
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {1,2,3,5}
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,2,3,5}
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,5,4,3,2,6] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 10
[[[.,.],[.,[[.,.],.]]],.]
=> [1,4,5,3,2,6] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 10
[[[.,.],[[.,.],[.,.]]],.]
=> [1,3,5,4,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 8
[[[.,.],[[.,[.,.]],.]],.]
=> [1,4,3,5,2,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
[[[.,.],[[[.,.],.],.]],.]
=> [1,3,4,5,2,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[[[[.,.],.],[.,[.,.]]],.]
=> [1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[[[[.,.],.],[[.,.],.]],.]
=> [1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[[[[.,.],[.,[.,.]]],.],.]
=> [1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[[[[.,.],[[.,.],.]],.],.]
=> [1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[[[[[.,.],.],[.,.]],.],.]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => [1,4,5,3,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> 13
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,5,6,4,3,2,7] => [1,4,6,2,5,3,7] => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> 10
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,4,6,5,3,2,7] => [1,6,2,4,5,3,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 13
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,5,4,6,3,2,7] => [1,6,2,5,3,4,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 13
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,4,5,6,3,2,7] => [1,5,3,6,2,4,7] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 11
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> 13
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,3,5,6,4,2,7] => [1,6,2,3,5,4,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> 12
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [1,4,3,6,5,2,7] => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 9
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 10
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,4,5,3,6,2,7] => [1,6,2,4,3,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> 11
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,3,5,4,6,2,7] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 9
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,4,3,5,6,2,7] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 9
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,2,6,5,4,3,7] => [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> 11
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,2,5,6,4,3,7] => [1,2,6,3,5,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> 11
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 9
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,2,5,4,6,3,7] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 8
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 9
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 8
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> 11
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,4,5,3,2,6,7] => [1,5,2,4,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> 11
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,3,5,4,2,6,7] => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> 9
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,4,3,5,2,6,7] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 8
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 9
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 8
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 8
Description
The 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.
Matching statistic: St001880
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 9%●distinct values known / distinct values provided: 8%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 9%●distinct values known / distinct values provided: 8%
Values
[.,.]
=> [1] => [1] => ([],1)
=> ? = 1
[.,[.,.]]
=> [2,1] => [2,1] => ([],2)
=> ? ∊ {1,2}
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,2}
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,2,2,5}
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,2,2,5}
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {1,2,2,5}
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,2,2,5}
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,5,4,3,2,6] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[[[.,.],[.,[[.,.],.]]],.]
=> [1,4,5,3,2,6] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 2
[[[.,.],[[.,.],[.,.]]],.]
=> [1,3,5,4,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[[[.,.],[[.,[.,.]],.]],.]
=> [1,4,3,5,2,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[[[.,.],[[[.,.],.],.]],.]
=> [1,3,4,5,2,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[[[[.,.],.],[.,[.,.]]],.]
=> [1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[[[[.,.],.],[[.,.],.]],.]
=> [1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[[[[.,.],[.,[.,.]]],.],.]
=> [1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[[[[.,.],[[.,.],.]],.],.]
=> [1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[[[[[.,.],.],[.,.]],.],.]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => [1,4,5,3,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> 2
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,5,6,4,3,2,7] => [1,4,6,2,5,3,7] => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> 4
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,4,6,5,3,2,7] => [1,6,2,4,5,3,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 2
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,5,4,6,3,2,7] => [1,6,2,5,3,4,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 2
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,4,5,6,3,2,7] => [1,5,3,6,2,4,7] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 4
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> 2
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,3,5,6,4,2,7] => [1,6,2,3,5,4,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> 2
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [1,4,3,6,5,2,7] => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 6
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 4
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,4,5,3,6,2,7] => [1,6,2,4,3,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> 4
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,3,5,4,6,2,7] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 6
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,4,3,5,6,2,7] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 6
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,2,6,5,4,3,7] => [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> 3
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,2,5,6,4,3,7] => [1,2,6,3,5,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> 3
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 5
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,2,5,4,6,3,7] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> 3
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,4,5,3,2,6,7] => [1,5,2,4,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> 3
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,3,5,4,2,6,7] => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> 5
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,4,3,5,2,6,7] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 5
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 9%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 9%
Values
[.,.]
=> [1] => [1] => ([],1)
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],.]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,5} - 1
[[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {2,5} - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14} - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42} - 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,6,4,3,2,1] => ([(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)
=> ? ∊ {2,2,2,2,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => [6,4,5,3,2,1] => ([(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)
=> ? ∊ {2,2,2,2,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [5,4,6,3,2,1] => ([(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)
=> ? ∊ {2,2,2,2,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => [6,5,3,4,2,1] => ([(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)
=> ? ∊ {2,2,2,2,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132} - 1
[[[.,.],.],[[[.,.],.],.]]
=> [1,2,4,5,6,3] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[[.,[.,[[[.,.],.],.]]],.]
=> [3,4,5,2,1,6] => [3,4,5,2,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[.,[[[.,.],.],[.,.]]],.]
=> [2,3,5,4,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[.,[[[[.,.],.],.],.]],.]
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,2,5,4,3,6] => [5,4,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[[.,[.,[.,[.,.]]]],.],.]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[[[.,[.,[.,.]]],.],.],.]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[[[[.,[.,.]],.],.],.],.]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> 1 = 2 - 1
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> 0 = 1 - 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 7 - 1
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 3 - 1
[[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[[.,[.,[[[.,.],.],.]]],.],.]
=> [3,4,5,2,1,6,7] => [3,4,5,2,1,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => [5,2,3,4,1,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[[.,[[[[.,.],.],.],.]],.],.]
=> [2,3,4,5,1,6,7] => [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(5,6)],7)
=> 1 = 2 - 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> 0 = 1 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001645
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Values
[.,.]
=> [1] => ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,5}
[[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,5}
[[[.,.],.],.]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,4,5,5,7,7,9,14}
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,9,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => ([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => ([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => ([(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)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,12,12,12,12,14,14,14,14,14,14,14,14,14,15,15,15,15,15,16,16,18,18,18,18,19,19,19,19,20,20,20,20,23,23,23,23,24,24,24,25,27,27,28,28,28,28,30,30,30,30,34,34,34,37,37,42,42,43,43,43,43,48,56,56,57,66,66,76,76,90,132}
[.,[.,[[[[.,.],.],.],.]]]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ([],7)
=> ([],1)
=> 1
Description
The pebbling number of a connected graph.
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