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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St000280
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
St000280: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 1
[1,2,3] => 5
[1,3,2] => 3
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 14
[1,2,4,3] => 9
[1,3,2,4] => 7
[1,3,4,2] => 9
[1,4,2,3] => 7
[1,4,3,2] => 4
[2,1,3,4] => 5
[2,1,4,3] => 3
[2,3,1,4] => 7
[2,3,4,1] => 9
[2,4,1,3] => 7
[2,4,3,1] => 4
[3,1,2,4] => 5
[3,1,4,2] => 3
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 7
[3,4,2,1] => 4
[4,1,2,3] => 5
[4,1,3,2] => 3
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 42
[1,2,3,5,4] => 28
[1,2,4,3,5] => 23
[1,2,4,5,3] => 28
[1,2,5,3,4] => 23
[1,2,5,4,3] => 14
[1,3,2,4,5] => 19
[1,3,2,5,4] => 12
[1,3,4,2,5] => 23
[1,3,4,5,2] => 28
[1,3,5,2,4] => 23
[1,3,5,4,2] => 14
[1,4,2,3,5] => 19
[1,4,2,5,3] => 12
[1,4,3,2,5] => 9
[1,4,3,5,2] => 12
[1,4,5,2,3] => 23
[1,4,5,3,2] => 14
Description
The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations.
Matching statistic: St000032
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 5
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 14
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 7
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 7
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 9
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 7
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 7
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 9
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 7
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 9
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 42
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 14
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 19
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 14
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 19
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 23
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 7
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 19
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 14
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 19
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 23
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 7
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 9
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 7
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 19
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
Description
The number of elements smaller than the given Dyck path in the Tamari Order.
Matching statistic: St000082
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000082: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000082: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [[.,.],.]
=> 1
[2,1] => [[.,.],.]
=> [1,2] => [.,[.,.]]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [[.,.],[.,.]]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [.,[[.,.],.]]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => [.,[[.,.],.]]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> 5
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 5
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 7
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 7
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 7
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 3
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 9
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 9
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 3
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 9
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 5
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 7
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 7
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 5
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 14
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 5
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> 7
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> 7
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> 7
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 9
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 9
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> 3
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 9
Description
The number of elements smaller than a binary tree in Tamari order.
Matching statistic: St000420
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 5
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 9
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 9
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 9
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 14
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 14
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 12
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 14
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 12
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 9
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 12
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 14
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 13 = 14 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 11 = 12 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 11 = 12 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 8 = 9 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 11 = 12 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St001232
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 21%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 21%
Values
[1,2] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 5 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,7,7,7,7,7,9,9,9,14} - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,4,3,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,5,1,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,5,3,1,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,5,4,1,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,1,4,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,1,5,2,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,1,5,4,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,2,4,1,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[3,5,1,2,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[4,1,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[4,5,1,2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[5,1,2,3,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,6,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,3,5,4,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,2,3,5,6,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [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 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 21%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 21%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [2,1] => [1,2] => ([],2)
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,5} - 1
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 1 = 2 - 1
[2,3,1] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,5} - 1
[3,2,1] => [3,2,1] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[1,3,4,2] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,3,4,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,4,3,1] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,1,4,2] => [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[3,2,4,1] => [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,1,2] => [2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,4,2,1] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,2,3,1] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,2,4,5,3] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,4,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,4,5,2] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,5,4,2] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,2,5,3] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,3,5,2] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,5,3,2] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,3,4,2] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,4,3,2] => [1,5,4,3,2] => [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
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,4,5,3] => [2,1,3,5,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,3,1,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,1,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,4,1,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,4,5,1] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,5,1,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,5,4,1] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,4,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[2,5,4,3,1] => [1,5,4,3,2] => [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
[3,1,5,4,2] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,5,4,1] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,5,4,2,1] => [1,5,4,3,2] => [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
[4,1,5,3,2] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,2,1,5,3] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,2,5,3,1] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,1,5,2] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,3,2,5,1] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,3,5,2,1] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,5,3,2,1] => [1,5,4,3,2] => [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
[5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,6,5,4,3,2] => [1,6,5,4,3,2] => [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
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,6,3,4,5] => [2,1,6,3,4,5] => [1,4,5,6,3,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,1,6,5,4,3] => [2,1,6,5,4,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,6,5,4,3,1] => [1,6,5,4,3,2] => [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
[3,1,6,5,4,2] => [2,1,6,5,4,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[3,2,1,6,5,4] => [3,2,1,6,5,4] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,2,6,1,4,5] => [3,2,6,1,4,5] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[3,2,6,5,4,1] => [2,1,6,5,4,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[3,6,5,4,2,1] => [1,6,5,4,3,2] => [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
[4,1,6,5,3,2] => [2,1,6,5,4,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,2,1,6,5,3] => [3,2,1,6,5,4] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[4,2,6,5,3,1] => [2,1,6,5,4,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,3,1,6,5,2] => [3,2,1,6,5,4] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[4,3,2,1,5,6] => [4,3,2,1,5,6] => [3,2,1,5,6,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> 2 = 3 - 1
[4,3,2,1,6,5] => [4,3,2,1,6,5] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 3 - 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: St001880
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 17%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 17%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,2}
[2,1] => [2,1] => [2,1] => ([],2)
=> ? ∊ {1,2}
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {1,2,2,3,5}
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,2,2,3,5}
[2,3,1] => [3,2,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,2,2,3,5}
[3,1,2] => [3,2,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,2,2,3,5}
[3,2,1] => [3,2,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,2,2,3,5}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => ([],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[2,4,3,1] => [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[3,1,4,2] => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[3,2,4,1] => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[3,4,1,2] => [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[3,4,2,1] => [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[4,1,2,3] => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[4,1,3,2] => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[4,2,1,3] => [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[4,2,3,1] => [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[4,3,1,2] => [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[4,3,2,1] => [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,5,5,5,7,7,7,7,7,9,9,9,14}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,5,3,2] => [1,5,4,3,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,2,3,4] => [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,2,4,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,3,2,4] => [1,5,4,3,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,3,4,2] => [1,5,4,3,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,4,2,3] => [1,5,4,3,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,4,3,2] => [1,5,4,3,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,5,4,6] => [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
[1,2,4,3,5,6] => [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
[1,2,4,5,3,6] => [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
[1,2,5,3,4,6] => [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
[1,2,5,4,3,6] => [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
[1,3,2,4,5,6] => [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
[1,3,4,2,5,6] => [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
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[1,4,2,3,5,6] => [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
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,4,3,2,5,6] => [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
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001879: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 10%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001879: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 10%
Values
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ? ∊ {1,2}
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,2}
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? ∊ {1,2,2,5}
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,5}
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,5}
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,2,2,5}
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,2,2,2,3,4,4,5,5,5,7,7,7,7,9,9,9,14}
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42}
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
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: St001877
Values
[1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 2 - 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 2 - 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2 = 3 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 3 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 3 - 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {4,4,4,5,5,5,7,7,7,7,7,9,9,9,14} - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 1 = 2 - 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 2 - 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 2 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 2 - 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,28,28,28,28,42} - 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
Description
Number of indecomposable injective modules with projective dimension 2.
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St001633The number of simple modules with projective dimension two in the incidence algebra of the poset.
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