Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001384
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 0
[1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 0
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 0
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St001879
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2,3] => [.,[.,[.,.]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[1,3,2] => [.,[[.,.],.]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 2 = 0 + 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 1 + 2
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 1 + 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 0 + 2
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 0 + 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 2 + 2
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 2 + 2
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 2 + 2
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 2 + 2
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,2,3,5,4] => [.,[[.,[.,[[.,.],.]]],.]]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
 => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,3,4,6,5] => [.,[[.,[.,[.,[[.,.],.]]]],.]]
 => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],.]]
 => [[.,[.,[.,[[.,[.,.]],.]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],.]]
 => [[.,[.,[.,[[[.,.],.],.]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,6,3,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],.]]
 => [[.,[.,[[.,[.,[.,.]]],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,6,3,5,4] => [.,[[.,[[.,[[.,.],.]],.]],.]]
 => [[.,[.,[[.,[[.,.],.]],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,6,5,3,4] => [.,[[.,[[[.,[.,.]],.],.]],.]]
 => [[.,[.,[[[.,[.,.]],.],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,6,5,4,3] => [.,[[.,[[[[.,.],.],.],.]],.]]
 => [[.,[.,[[[[.,.],.],.],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,2,3,4,5] => [.,[[[.,[.,[.,[.,.]]]],.],.]]
 => [[.,[[.,[.,[.,[.,.]]]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,2,3,5,4] => [.,[[[.,[.,[[.,.],.]]],.],.]]
 => [[.,[[.,[.,[[.,.],.]]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,2,5,3,4] => [.,[[[.,[[.,[.,.]],.]],.],.]]
 => [[.,[[.,[[.,[.,.]],.]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,2,5,4,3] => [.,[[[.,[[[.,.],.],.]],.],.]]
 => [[.,[[.,[[[.,.],.],.]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,5,2,3,4] => [.,[[[[.,[.,[.,.]]],.],.],.]]
 => [[.,[[[.,[.,[.,.]]],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,5,2,4,3] => [.,[[[[.,[[.,.],.]],.],.],.]]
 => [[.,[[[.,[[.,.],.]],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,5,4,2,3] => [.,[[[[[.,[.,.]],.],.],.],.]]
 => [[.,[[[[.,[.,.]],.],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => [[.,[[[[[.,.],.],.],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2,3] => [.,[.,[.,.]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 3
[1,3,2] => [.,[[.,.],.]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 0 + 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 1 + 3
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 1 + 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 0 + 3
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 0 + 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 2 + 3
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 2 + 3
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 3
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 3
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 2 + 3
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 2 + 3
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 3
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 3
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 3
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 3
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,2,3,5,4] => [.,[[.,[.,[[.,.],.]]],.]]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
 => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,3,4,6,5] => [.,[[.,[.,[.,[[.,.],.]]]],.]]
 => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],.]]
 => [[.,[.,[.,[[.,[.,.]],.]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],.]]
 => [[.,[.,[.,[[[.,.],.],.]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,6,3,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],.]]
 => [[.,[.,[[.,[.,[.,.]]],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,6,3,5,4] => [.,[[.,[[.,[[.,.],.]],.]],.]]
 => [[.,[.,[[.,[[.,.],.]],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,6,5,3,4] => [.,[[.,[[[.,[.,.]],.],.]],.]]
 => [[.,[.,[[[.,[.,.]],.],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,6,5,4,3] => [.,[[.,[[[[.,.],.],.],.]],.]]
 => [[.,[.,[[[[.,.],.],.],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,2,3,4,5] => [.,[[[.,[.,[.,[.,.]]]],.],.]]
 => [[.,[[.,[.,[.,[.,.]]]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,2,3,5,4] => [.,[[[.,[.,[[.,.],.]]],.],.]]
 => [[.,[[.,[.,[[.,.],.]]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,2,5,3,4] => [.,[[[.,[[.,[.,.]],.]],.],.]]
 => [[.,[[.,[[.,[.,.]],.]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,2,5,4,3] => [.,[[[.,[[[.,.],.],.]],.],.]]
 => [[.,[[.,[[[.,.],.],.]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,5,2,3,4] => [.,[[[[.,[.,[.,.]]],.],.],.]]
 => [[.,[[[.,[.,[.,.]]],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,5,2,4,3] => [.,[[[[.,[[.,.],.]],.],.],.]]
 => [[.,[[[.,[[.,.],.]],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,5,4,2,3] => [.,[[[[[.,[.,.]],.],.],.],.]]
 => [[.,[[[[.,[.,.]],.],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => [[.,[[[[[.,.],.],.],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.