Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001004
(load all 4 compositions to match this statistic)
Mapping
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 2
[1,2,3] => 3
[1,3,2] => 3
[2,1,3] => 3
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 2
[1,2,3,4] => 4
[1,2,4,3] => 4
[1,3,2,4] => 4
[1,3,4,2] => 4
[1,4,2,3] => 4
[1,4,3,2] => 3
[2,1,3,4] => 4
[2,1,4,3] => 4
[2,3,1,4] => 4
[2,3,4,1] => 4
[2,4,1,3] => 4
[2,4,3,1] => 3
[3,1,2,4] => 4
[3,1,4,2] => 4
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 3
[4,1,2,3] => 4
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 2
[4,3,1,2] => 3
[4,3,2,1] => 2
[1,2,3,4,5] => 5
[1,2,3,5,4] => 5
[1,2,4,3,5] => 5
[1,2,4,5,3] => 5
[1,2,5,3,4] => 5
[1,2,5,4,3] => 4
[1,3,2,4,5] => 5
[1,3,2,5,4] => 5
[1,3,4,2,5] => 5
[1,3,4,5,2] => 5
[1,3,5,2,4] => 5
[1,3,5,4,2] => 4
[1,4,2,3,5] => 5
[1,4,2,5,3] => 5
[1,4,3,2,5] => 4
[1,4,3,5,2] => 4
[1,4,5,2,3] => 5
Description
The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St001880
(load all 5 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Values
[1] => [1,0]
 => [.,.]
 => ([],1)
 => ? = 1
[1,2] => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2
[2,1] => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 2
[1,2,3] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3
[2,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3
[2,3,1] => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3
[3,1,2] => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 3
[3,2,1] => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 4
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 5 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Values
[1] => [1,0]
 => [.,.]
 => ([],1)
 => ? = 1 - 1
[1,2] => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2 - 1
[2,1] => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 4 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 4 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5 - 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 - 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 - 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5 - 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 - 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5 - 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 - 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5 - 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5 - 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5 - 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 5 - 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
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: St001876
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
[1] => [1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1 - 1
[1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,2,3] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 3 - 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 3 - 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 3 - 1
[2,3,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 3 - 1
[3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 3 - 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,2,3,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,2,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,3,2,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1)],2)
 => ? = 3 - 1
[2,1,3,4] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => ? = 4 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => ? = 4 - 1
[2,3,1,4] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 1
[2,3,4,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 4 - 1
[2,4,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 1
[2,4,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 3 - 1
[3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 1
[3,1,4,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1)],2)
 => ? = 3 - 1
[3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 3 - 1
[3,4,1,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => ? = 4 - 1
[3,4,2,1] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1)],2)
 => ? = 3 - 1
[4,1,2,3] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 4 - 1
[4,1,3,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 3 - 1
[4,2,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 3 - 1
[4,2,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,3,1,2] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1)],2)
 => ? = 3 - 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,2,3,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,2,3,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,2,4,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,2,4,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,2,5,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,2,5,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,3,2,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,3,2,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,3,4,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,3,4,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,3,5,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,4,2,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,4,2,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,4,3,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,5,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 5 - 1
[1,5,2,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 4 - 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 3 - 1
[5,2,3,4,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[5,3,2,4,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[5,3,4,2,1] => [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[5,4,2,3,1] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,2,3,4,5,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,2,3,5,4,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,2,4,3,5,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,2,4,5,3,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,2,5,3,4,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,2,5,4,3,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,3,2,4,5,1] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,3,2,5,4,1] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,3,4,2,5,1] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,3,4,5,2,1] => [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,3,5,2,4,1] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,3,5,4,2,1] => [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,4,2,3,5,1] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,4,2,5,3,1] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,4,3,2,5,1] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,4,3,5,2,1] => [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,4,5,2,3,1] => [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,4,5,3,2,1] => [6,4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,5,2,3,4,1] => [6,5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,5,2,4,3,1] => [6,5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,5,3,2,4,1] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,5,3,4,2,1] => [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[6,5,4,2,3,1] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,6,5,7,3,2,4,1] => [8,6,5,7,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,15),(4,15),(5,13),(6,12),(7,17),(8,18),(10,9),(11,9),(12,10),(13,11),(14,17),(15,18),(16,10),(16,11),(17,12),(17,16),(18,13),(18,16)],19)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,5,6,7,2,3,4,1] => [8,5,7,6,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,15),(4,15),(5,13),(6,12),(7,17),(8,18),(10,9),(11,9),(12,10),(13,11),(14,17),(15,18),(16,10),(16,11),(17,12),(17,16),(18,13),(18,16)],19)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,3,2,5,4,7,6,1] => [8,3,2,7,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,13),(3,16),(4,15),(5,17),(6,17),(7,15),(7,19),(8,16),(8,19),(10,12),(11,12),(12,18),(13,9),(14,9),(15,10),(16,11),(17,18),(18,13),(18,14),(19,10),(19,11)],20)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,3,2,7,6,5,4,1] => [8,3,2,7,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,13),(3,16),(4,15),(5,17),(6,17),(7,15),(7,19),(8,16),(8,19),(10,12),(11,12),(12,18),(13,9),(14,9),(15,10),(16,11),(17,18),(18,13),(18,14),(19,10),(19,11)],20)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,5,4,3,2,7,6,1] => [8,5,4,3,2,7,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,13),(3,16),(4,15),(5,17),(6,17),(7,15),(7,19),(8,16),(8,19),(10,12),(11,12),(12,18),(13,9),(14,9),(15,10),(16,11),(17,18),(18,13),(18,14),(19,10),(19,11)],20)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,4,5,2,3,7,6,1] => [8,4,7,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,6,4,3,7,2,5,1] => [8,6,4,3,7,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,4,6,2,7,3,5,1] => [8,4,7,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,3,2,6,7,4,5,1] => [8,3,2,7,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,13),(3,16),(4,15),(5,17),(6,17),(7,15),(7,19),(8,16),(8,19),(10,12),(11,12),(12,18),(13,9),(14,9),(15,10),(16,11),(17,18),(18,13),(18,14),(19,10),(19,11)],20)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,4,7,2,6,5,3,1] => [8,4,7,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,5,7,6,2,4,3,1] => [8,5,7,6,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,15),(4,15),(5,13),(6,12),(7,17),(8,18),(10,9),(11,9),(12,10),(13,11),(14,17),(15,18),(16,10),(16,11),(17,12),(17,16),(18,13),(18,16)],19)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[8,5,3,6,2,7,4,1] => [8,5,3,7,2,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001875
(load all 6 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 38%
Values
[1] => ([],1)
 => ([(0,1)],2)
 => ? = 1 + 1
[1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4 = 3 + 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4 = 3 + 1
[2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 3 + 1
[3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 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)
 => ? = 2 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 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)
 => 5 = 4 + 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)
 => 5 = 4 + 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)
 => 5 = 4 + 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)
 => 5 = 4 + 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)
 => ? = 3 + 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)
 => 5 = 4 + 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)
 => 5 = 4 + 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)
 => 5 = 4 + 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 + 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 + 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)
 => ? = 3 + 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)
 => 5 = 4 + 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 + 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)
 => ? = 3 + 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)
 => ? = 3 + 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 + 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)
 => ? = 3 + 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 + 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)
 => ? = 3 + 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)
 => ? = 3 + 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)
 => ? = 2 + 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)
 => ? = 3 + 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)
 => ? = 2 + 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)
 => 6 = 5 + 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)
 => 6 = 5 + 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)
 => 6 = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 4 + 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)
 => 6 = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 4 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 4 + 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)
 => ? = 4 + 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)
 => ? = 5 + 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)
 => ? = 4 + 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)
 => ? = 5 + 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)
 => ? = 4 + 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)
 => ? = 4 + 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 + 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)
 => ? = 4 + 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 + 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)
 => 6 = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 4 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => ? = 4 + 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)
 => ? = 5 + 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)
 => ? = 5 + 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)
 => 7 = 6 + 1
Description
The number of simple modules with projective dimension at most 1.