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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000082
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000082: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> 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
Description
The number of elements smaller than a binary tree in Tamari order.
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
[.,[.,.]]
=> [1,1,0,0]
=> 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
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 14
Description
The number of elements smaller than the given Dyck path in the 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: 17% ●values known / values provided: 21%●distinct values known / distinct values provided: 17%
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: 17% ●values known / values provided: 21%●distinct values known / distinct values provided: 17%
Values
[.,[.,.]]
=> [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
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 17%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 17%
Values
[.,[.,.]]
=> [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
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: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 17%
Values
[.,[.,.]]
=> [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
Description
The pebbling number of a connected graph.
Matching statistic: St001879
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 31%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 31%
Values
[.,[.,.]]
=> [2,1] => [2,1] => ([],2)
=> ? ∊ {1,2}
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,2}
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => ([],3)
=> ? ∊ {1,2,3,5}
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,2,3,5}
[[.,.],[.,.]]
=> [3,1,2] => [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] => [4,3,2,1] => ([],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => ([(1,3),(2,3)],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}
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,4,5,5,7,7,9,14}
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [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] => [3,2,1,4] => ([(0,3),(1,3),(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}
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [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] => [5,4,3,2,1] => ([],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,7,7,7,7,9,9,9,9,10,10,12,12,14,14,14,16,19,19,23,23,28,42}
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [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
[[[.,.],[.,[.,[.,.]]]],.]
=> [5,4,3,1,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 16
[[[.,.],[.,[[.,.],.]]],.]
=> [4,5,3,1,2,6] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 12
[[[.,.],[[.,.],[.,.]]],.]
=> [5,3,4,1,2,6] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 10
[[[.,.],[[.,[.,.]],.]],.]
=> [4,3,5,1,2,6] => [1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 10
[[[.,.],[[[.,.],.],.]],.]
=> [3,4,5,1,2,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[[[[.,.],.],[.,[.,.]]],.]
=> [5,4,1,2,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 10
[[[[.,.],.],[[.,.],.]],.]
=> [4,5,1,2,3,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
[[[[[.,.],.],.],[.,.]],.]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[[[[.,.],[.,[.,.]]],.],.]
=> [4,3,1,2,5,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 10
[[[[.,.],[[.,.],.]],.],.]
=> [3,4,1,2,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[[[[[.,.],.],[.,.]],.],.]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => [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
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)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 11%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 11%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [2,1] => ([],2)
=> ? ∊ {1,2}
[[.,.],.]
=> [1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> ? ∊ {1,2}
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,2,2,5}
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,2,2,5}
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {1,2,2,5}
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,2,2,5}
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [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}
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,2,2,2,3,3,5,5,7,7,9,14}
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [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}
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [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}
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [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}
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(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}
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(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}
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [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}
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [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}
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(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}
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(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}
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(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}
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(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}
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(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}
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(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}
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(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}
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [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}
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [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}
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [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}
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(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}
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(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}
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [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}
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [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}
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [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}
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [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}
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [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}
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [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}
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [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}
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [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}
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [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}
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(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}
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [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}
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(0,3),(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}
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => ([(0,3),(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}
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[[[.,.],[.,[[.,.],.]]],.]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[[[.,.],[[.,.],[.,.]]],.]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[[[.,.],[[.,[.,.]],.]],.]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[[[.,.],[[[.,.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[[[[.,.],.],[[.,.],.]],.]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[[[[.,.],[.,[.,.]]],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[[[[.,.],[[.,.],.]],.],.]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[[[[[.,.],.],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[[[[[[.,.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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