- FindStat

Your data matches 345 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001232

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St000781
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 43%●distinct values known / distinct values provided: 40%

Values

[.,.]
 => ([],1)
 => [2]
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => [3]
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => [3]
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [4]
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [4]
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => ? = 3
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => ? = 3
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8,2,2]
 => ? = 5
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8,2,2]
 => ? = 5
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,4,2,2]
 => 5
[[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[.,.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,4,2,2]
 => 5
[[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,4,2,2]
 => 5
[[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[[.,[.,.]],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8,2,2]
 => ? = 5
[[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8,2,2]
 => ? = 5
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [9,2,2]
 => ? = 7
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [9,2,2]
 => ? = 7
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [6,4,2,2]
 => ? = 7
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [6,4,2,2]
 => ? = 7
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [6,4,2,2]
 => ? = 7
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [6,4,2,2]
 => ? = 7
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [9,2,2]
 => ? = 7
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [9,2,2]
 => ? = 7
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [11,2,2]
 => ? = 7
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [11,2,2]
 => ? = 7
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [15,2,2,2,2]
 => ? = 4
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,.],[[[.,.],.],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [15,2,2,2,2]
 => ? = 4
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[[[.,.],.],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [11,2,2]
 => ? = 7
[[.,.],[[[.,.],[.,[.,.]]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[[[.,.],[[.,.],.]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[[[.,[.,.]],[.,.]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7

Description

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

Matching statistic: St000527

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 41%●distinct values known / distinct values provided: 40%

Values

[.,.]
 => ([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => 3
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => 3
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 5
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 5
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? = 5
[[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 5
[[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 5
[[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 5
[[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 5
[[.,.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? = 5
[[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? = 5
[[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 5
[[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 5
[[[.,[.,.]],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 5
[[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 5
[[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? = 5
[[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 5
[[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 5
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
 => ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
 => ? = 7
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
 => ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
 => ? = 7
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
 => ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
 => ? = 7
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ? = 7
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ? = 7
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ? = 7
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ? = 7
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
 => ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
 => ? = 7
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
 => ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
 => ? = 7
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ? = 7
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ? = 7
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ? = 7
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,11),(4,8),(4,9),(5,9),(5,10),(6,2),(7,1),(7,8),(7,10),(8,12),(8,15),(9,15),(10,13),(10,15),(11,6),(12,14),(13,3),(13,14),(14,11),(15,14)],16)
 => ? = 7
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
 => ([(0,3),(0,4),(0,5),(1,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(7,1),(7,11),(8,13),(9,13),(10,7),(10,13),(11,12),(12,6),(13,11)],14)
 => ? = 7
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
 => ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
 => ? = 7
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
 => ([(0,3),(0,4),(0,5),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,6),(8,12),(9,12),(10,7),(11,2),(11,12),(12,10)],13)
 => ? = 7
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
 => ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
 => ? = 7
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ? = 7
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ? = 7
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ? = 7
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ? = 7
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
 => ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
 => ? = 7
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ? = 7
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ? = 7
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => ?
 => ?
 => ? = 4
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ? = 7
[[.,.],[[.,.],[[[.,.],.],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ? = 7
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ? = 7
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ? = 7
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => ?
 => ?
 => ? = 4
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ? = 7
[[.,.],[[[[.,.],.],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ([(0,5),(0,6),(0,7),(2,10),(3,8),(3,9),(4,3),(4,12),(4,13),(5,11),(5,14),(6,11),(6,15),(7,4),(7,14),(7,15),(8,18),(9,2),(9,18),(10,1),(11,16),(12,8),(12,17),(13,9),(13,17),(14,12),(14,16),(15,13),(15,16),(16,17),(17,18),(18,10)],19)
 => ? = 7
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
 => ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
 => ? = 7
[[.,.],[[[.,.],[.,[.,.]]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ([(0,4),(0,5),(0,7),(1,12),(1,13),(3,14),(4,8),(4,10),(5,9),(5,10),(6,3),(6,11),(7,1),(7,8),(7,9),(8,13),(8,16),(9,12),(9,16),(10,16),(11,14),(12,15),(13,6),(13,15),(14,2),(15,11),(16,15)],17)
 => ? = 7

Description

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.

Matching statistic: St001597
(load all 4 compositions to match this statistic)

Mapping

Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 40% ●values known / values provided: 41%●distinct values known / distinct values provided: 40%

Values

[.,.]
 => [1,0]
 => [[1],[]]
 => 1
[.,[.,.]]
 => [1,0,1,0]
 => [[1,1],[]]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [[2],[]]
 => 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => 1
[[[.,.],.],.]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => 1
[[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 3
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => 1
[[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1,1],[]]
 => 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1,1],[]]
 => 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1],[]]
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ? = 5
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ? = 5
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => 1
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => 1
[.,[[[.,[[.,.],.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => 1
[.,[[[[.,[.,.]],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => 1
[.,[[[[[.,.],.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => 1
[[.,.],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ? = 5
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ? = 5
[[.,.],[[.,.],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[4,3,2],[1]]
 => ? = 5
[[.,.],[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1]]
 => ? = 5
[[.,.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2],[1]]
 => ? = 5
[[.,.],[[[.,.],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[4,4,2],[2]]
 => ? = 5
[[.,[[.,.],[.,.]]],[.,.]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [[3,3,3,2],[]]
 => ? = 5
[[[.,.],[.,[.,.]]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ? = 5
[[[.,.],[[.,.],.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [[4,3,3],[]]
 => ? = 5
[[[.,[.,.]],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [[3,3,3,3],[1]]
 => ? = 5
[[[[.,.],.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[4,4,3],[]]
 => ? = 5
[[[[.,.],[.,.]],.],[.,.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [[4,4,4],[1]]
 => ? = 5
[[.,[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[2,2,2,2,2],[1,1,1,1]]
 => 1
[[.,[.,[.,[[.,.],.]]]],.]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[3,2,2,2],[1,1,1]]
 => 1
[[[.,.],[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3],[2,1]]
 => ? = 5
[[[[.,.],[.,.]],[.,.]],.]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 5
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1,1],[1]]
 => ? = 7
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1,1],[]]
 => ? = 7
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2,1],[1,1]]
 => ? = 7
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2,1],[2,1]]
 => ? = 7
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [[4,3,2,1],[1]]
 => ? = 7
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [[3,3,3,2,1],[1,1]]
 => ? = 7
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2,1],[1]]
 => ? = 7
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [[4,4,2,1],[2]]
 => ? = 7
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [[3,3,3,2,1],[]]
 => ? = 7
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,1],[2]]
 => ? = 7
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [[4,3,3,1],[]]
 => ? = 7
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [[3,3,3,3,1],[1]]
 => ? = 7
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [[4,4,3,1],[]]
 => ? = 7
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [[4,4,4,1],[1]]
 => ? = 7
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3,1],[2,1]]
 => ? = 7
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,1],[1,1]]
 => ? = 7
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [[3,3,2,2,2],[1,1,1]]
 => ? = 7
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2,2],[2,1,1]]
 => ? = 7
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2,2],[1,1]]
 => ? = 7
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [[3,3,3,2,2],[1,1,1]]
 => ? = 7
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [[4,4,2,2],[1,1]]
 => ? = 7
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2,2],[2,1]]
 => ? = 7
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => [[3,3,3,3,2],[2,2,1]]
 => ? = 7
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3,2],[2,1]]
 => ? = 7
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3,2],[2,1]]
 => ? = 4
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3,2],[3,1]]
 => ? = 7
[[.,.],[[.,.],[[[.,.],.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [[5,3,2],[1]]
 => ? = 7
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [[3,3,3,3,2],[1,1,1]]
 => ? = 7
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [[4,4,3,2],[1,1]]
 => ? = 7
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1]]
 => ? = 4
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4,2],[2,1]]
 => ? = 7
[[.,.],[[[[.,.],.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [[5,5,2],[1]]
 => ? = 7
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [[4,4,3,2],[2,2]]
 => ? = 7

Description

The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.

Matching statistic: St000632
(load all 4 compositions to match this statistic)

Mapping

Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 41%●distinct values known / distinct values provided: 40%

Values

[.,.]
 => [1,0]
 => ([],1)
 => 0 = 1 - 1
[.,[.,.]]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,.],.]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(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)
 => ? = 3 - 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 5 - 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(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
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[[[.,.],.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[.,.],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 5 - 1
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 5 - 1
[[.,.],[[.,.],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 5 - 1
[[.,.],[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(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
[[.,.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(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
[[.,.],[[[.,.],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 5 - 1
[[.,[[.,.],[.,.]]],[.,.]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 5 - 1
[[[.,.],[.,[.,.]]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 5 - 1
[[[.,.],[[.,.],.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 5 - 1
[[[.,[.,.]],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 5 - 1
[[[[.,.],.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 5 - 1
[[[[.,.],[.,.]],.],[.,.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 5 - 1
[[.,[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[.,[.,[.,[[.,.],.]]]],.]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[[.,.],[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 5 - 1
[[[[.,.],[.,.]],[.,.]],.]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(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
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 7 - 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
 => ? = 7 - 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
 => ? = 7 - 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 7 - 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
 => ? = 7 - 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
 => ? = 7 - 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? = 7 - 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 7 - 1
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 7 - 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
 => ? = 7 - 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
 => ? = 7 - 1
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
 => ? = 4 - 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[[[.,.],.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,8),(3,10),(4,10),(5,6),(5,7),(6,2),(6,9),(7,9),(9,8),(10,1),(10,5)],11)
 => ? = 7 - 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
 => ? = 7 - 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,10),(2,9),(3,7),(4,7),(5,2),(5,8),(6,1),(6,8),(7,5),(7,6),(8,9),(8,10),(9,11),(10,11)],12)
 => ? = 4 - 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
 => ? = 7 - 1
[[.,.],[[[[.,.],.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,8),(3,10),(4,10),(5,6),(5,7),(6,2),(6,9),(7,9),(9,8),(10,1),(10,5)],11)
 => ? = 7 - 1
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1

Description

The jump number of the poset. A jump in a linear extension

$e_1, \dots, e_n$ of a poset

$P$ is a pair

$(e_i, e_{i+1})$ so that

$e_{i+1}$ does not cover

$e_i$ in

$P$ . The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

Matching statistic: St001397
(load all 4 compositions to match this statistic)

Mapping

Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 41%●distinct values known / distinct values provided: 40%

Values

[.,.]
 => [1,0]
 => ([],1)
 => 0 = 1 - 1
[.,[.,.]]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,.],.]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(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)
 => ? = 3 - 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 5 - 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(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
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[[[.,.],.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[.,.],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 5 - 1
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 5 - 1
[[.,.],[[.,.],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 5 - 1
[[.,.],[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(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
[[.,.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(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
[[.,.],[[[.,.],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 5 - 1
[[.,[[.,.],[.,.]]],[.,.]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 5 - 1
[[[.,.],[.,[.,.]]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 5 - 1
[[[.,.],[[.,.],.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 5 - 1
[[[.,[.,.]],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 5 - 1
[[[[.,.],.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 5 - 1
[[[[.,.],[.,.]],.],[.,.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 5 - 1
[[.,[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[.,[.,[.,[[.,.],.]]]],.]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[[.,.],[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 5 - 1
[[[[.,.],[.,.]],[.,.]],.]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(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
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 7 - 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
 => ? = 7 - 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
 => ? = 7 - 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 7 - 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
 => ? = 7 - 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
 => ? = 7 - 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? = 7 - 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 7 - 1
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 7 - 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
 => ? = 7 - 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
 => ? = 7 - 1
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
 => ? = 4 - 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[[[.,.],.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,8),(3,10),(4,10),(5,6),(5,7),(6,2),(6,9),(7,9),(9,8),(10,1),(10,5)],11)
 => ? = 7 - 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
 => ? = 7 - 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,10),(2,9),(3,7),(4,7),(5,2),(5,8),(6,1),(6,8),(7,5),(7,6),(8,9),(8,10),(9,11),(10,11)],12)
 => ? = 4 - 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
 => ? = 7 - 1
[[.,.],[[[[.,.],.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,8),(3,10),(4,10),(5,6),(5,7),(6,2),(6,9),(7,9),(9,8),(10,1),(10,5)],11)
 => ? = 7 - 1
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1

Description

Number of pairs of incomparable elements in a finite poset. For a finite poset

$(P,\leq)$ , this is the number of unordered pairs

$\{x,y\} \in \binom{P}{2}$ with

$x \not\leq y$ and

$y \not\leq x$ .

Matching statistic: St001596
(load all 4 compositions to match this statistic)

Mapping

Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 40% ●values known / values provided: 41%●distinct values known / distinct values provided: 40%

Values

[.,.]
 => [1,0]
 => [[1],[]]
 => 0 = 1 - 1
[.,[.,.]]
 => [1,0,1,0]
 => [[1,1],[]]
 => 0 = 1 - 1
[[.,.],.]
 => [1,1,0,0]
 => [[2],[]]
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 3 - 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => 0 = 1 - 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => 0 = 1 - 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => 0 = 1 - 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => 0 = 1 - 1
[[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 0 = 1 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => 0 = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1,1],[]]
 => 0 = 1 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ? = 5 - 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ? = 5 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => 0 = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => 0 = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => 0 = 1 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => 0 = 1 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => 0 = 1 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => 0 = 1 - 1
[.,[[[[[.,.],.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => 0 = 1 - 1
[[.,.],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ? = 5 - 1
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ? = 5 - 1
[[.,.],[[.,.],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[4,3,2],[1]]
 => ? = 5 - 1
[[.,.],[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1]]
 => ? = 5 - 1
[[.,.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2],[1]]
 => ? = 5 - 1
[[.,.],[[[.,.],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[4,4,2],[2]]
 => ? = 5 - 1
[[.,[[.,.],[.,.]]],[.,.]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [[3,3,3,2],[]]
 => ? = 5 - 1
[[[.,.],[.,[.,.]]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ? = 5 - 1
[[[.,.],[[.,.],.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [[4,3,3],[]]
 => ? = 5 - 1
[[[.,[.,.]],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [[3,3,3,3],[1]]
 => ? = 5 - 1
[[[[.,.],.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[4,4,3],[]]
 => ? = 5 - 1
[[[[.,.],[.,.]],.],[.,.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [[4,4,4],[1]]
 => ? = 5 - 1
[[.,[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[2,2,2,2,2],[1,1,1,1]]
 => 0 = 1 - 1
[[.,[.,[.,[[.,.],.]]]],.]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[3,2,2,2],[1,1,1]]
 => 0 = 1 - 1
[[[.,.],[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3],[2,1]]
 => ? = 5 - 1
[[[[.,.],[.,.]],[.,.]],.]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 5 - 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1,1],[1]]
 => ? = 7 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1,1],[]]
 => ? = 7 - 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2,1],[1,1]]
 => ? = 7 - 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2,1],[2,1]]
 => ? = 7 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [[4,3,2,1],[1]]
 => ? = 7 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [[3,3,3,2,1],[1,1]]
 => ? = 7 - 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2,1],[1]]
 => ? = 7 - 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [[4,4,2,1],[2]]
 => ? = 7 - 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [[3,3,3,2,1],[]]
 => ? = 7 - 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,1],[2]]
 => ? = 7 - 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [[4,3,3,1],[]]
 => ? = 7 - 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [[3,3,3,3,1],[1]]
 => ? = 7 - 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [[4,4,3,1],[]]
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [[4,4,4,1],[1]]
 => ? = 7 - 1
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3,1],[2,1]]
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,1],[1,1]]
 => ? = 7 - 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [[3,3,2,2,2],[1,1,1]]
 => ? = 7 - 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2,2],[2,1,1]]
 => ? = 7 - 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2,2],[1,1]]
 => ? = 7 - 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [[3,3,3,2,2],[1,1,1]]
 => ? = 7 - 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [[4,4,2,2],[1,1]]
 => ? = 7 - 1
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2,2],[2,1]]
 => ? = 7 - 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => [[3,3,3,3,2],[2,2,1]]
 => ? = 7 - 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3,2],[2,1]]
 => ? = 7 - 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3,2],[2,1]]
 => ? = 4 - 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3,2],[3,1]]
 => ? = 7 - 1
[[.,.],[[.,.],[[[.,.],.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [[5,3,2],[1]]
 => ? = 7 - 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [[3,3,3,3,2],[1,1,1]]
 => ? = 7 - 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [[4,4,3,2],[1,1]]
 => ? = 7 - 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1]]
 => ? = 4 - 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4,2],[2,1]]
 => ? = 7 - 1
[[.,.],[[[[.,.],.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [[5,5,2],[1]]
 => ? = 7 - 1
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [[4,4,3,2],[2,2]]
 => ? = 7 - 1

Description

The number of two-by-two squares inside a skew partition. This is, the number of cells

$(i,j)$ in a skew partition for which the box

$(i+1,j+1)$ is also a cell inside the skew partition.

Matching statistic: St001633
(load all 5 compositions to match this statistic)

Mapping

Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 41%●distinct values known / distinct values provided: 40%

Values

[.,.]
 => [1,0]
 => ([],1)
 => 0 = 1 - 1
[.,[.,.]]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,.],.]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(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)
 => ? = 3 - 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 5 - 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(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
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[.,[[[[[.,.],.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[.,.],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 5 - 1
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 5 - 1
[[.,.],[[.,.],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 5 - 1
[[.,.],[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(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
[[.,.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(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
[[.,.],[[[.,.],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 5 - 1
[[.,[[.,.],[.,.]]],[.,.]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 5 - 1
[[[.,.],[.,[.,.]]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 5 - 1
[[[.,.],[[.,.],.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 5 - 1
[[[.,[.,.]],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 5 - 1
[[[[.,.],.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 5 - 1
[[[[.,.],[.,.]],.],[.,.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 5 - 1
[[.,[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[.,[.,[.,[[.,.],.]]]],.]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[[.,.],[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 5 - 1
[[[[.,.],[.,.]],[.,.]],.]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(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
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 7 - 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
 => ? = 7 - 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
 => ? = 7 - 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 7 - 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
 => ? = 7 - 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
 => ? = 7 - 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? = 7 - 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 7 - 1
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 7 - 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
 => ? = 7 - 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
 => ? = 7 - 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
 => ? = 7 - 1
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
 => ? = 4 - 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,.],[[[.,.],.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ? = 7 - 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,8),(3,10),(4,10),(5,6),(5,7),(6,2),(6,9),(7,9),(9,8),(10,1),(10,5)],11)
 => ? = 7 - 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
 => ? = 7 - 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,10),(2,9),(3,7),(4,7),(5,2),(5,8),(6,1),(6,8),(7,5),(7,6),(8,9),(8,10),(9,11),(10,11)],12)
 => ? = 4 - 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
 => ? = 7 - 1
[[.,.],[[[[.,.],.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,8),(3,10),(4,10),(5,6),(5,7),(6,2),(6,9),(7,9),(9,8),(10,1),(10,5)],11)
 => ? = 7 - 1
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ? = 7 - 1

Description

The number of simple modules with projective dimension two in the incidence algebra of the poset.

Matching statistic: St001431
(load all 5 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 41%●distinct values known / distinct values provided: 40%

Values

[.,.]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 0 = 1 - 1
[[.,.],.]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[.,.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[[.,[.,.]],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5 - 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[.,.],[[[.,.],.],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[[[.,.],.],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1

Description

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I. See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.

Matching statistic: St000049
(load all 2 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000049: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 40%●distinct values known / distinct values provided: 20%

Values

[.,.]
 => ([],1)
 => [2]
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => [3]
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => [3]
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [4]
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [4]
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => ? = 3
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => ? = 3
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8,2,2]
 => ? = 5
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8,2,2]
 => ? = 5
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,4,2,2]
 => ? = 5
[[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[.,.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,4,2,2]
 => ? = 5
[[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,4,2,2]
 => ? = 5
[[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[[.,[.,.]],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 5
[[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,4,2,2]
 => ? = 5
[[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 1
[[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8,2,2]
 => ? = 5
[[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8,2,2]
 => ? = 5
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [9,2,2]
 => ? = 7
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [9,2,2]
 => ? = 7
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [6,4,2,2]
 => ? = 7
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [6,4,2,2]
 => ? = 7
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [6,4,2,2]
 => ? = 7
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[[.,[.,.]],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => [16]
 => ? = 7
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [6,4,2,2]
 => ? = 7
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [9,2,2]
 => ? = 7
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [9,2,2]
 => ? = 7
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [11,2,2]
 => ? = 7
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [9,8]
 => ? = 7
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [11,2,2]
 => ? = 7
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [15,2,2,2,2]
 => ? = 4
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,.],[[[.,.],.],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [15,2,2,2,2]
 => ? = 4
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7
[[.,.],[[[[.,.],.],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [11,4,4]
 => ? = 7

Description

The number of set partitions whose sorted block sizes correspond to the partition.

The following 335 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000256The number of parts from which one can substract 2 and still get an integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000307The number of rowmotion orbits of a poset. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001330The hat guessing number of a graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001618The cardinality of the Frattini sublattice of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001964The interval resolution global dimension of a poset. St000100The number of linear extensions of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000298The order dimension or Dushnik-Miller dimension of a poset. St000321The number of integer partitions of n that are dominated by an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000909The number of maximal chains of maximal size in a poset. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001763The Hurwitz number of an integer partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000644The number of graphs with given frequency partition. St000944The 3-degree of an integer partition. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000750The number of occurrences of the pattern 4213 in a permutation. St000741The Colin de Verdière graph invariant. St000406The number of occurrences of the pattern 3241 in a permutation. St001545The second Elser number of a connected graph. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000358The number of occurrences of the pattern 31-2. St000624The normalized sum of the minimal distances to a greater element. St000779The tier of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000068The number of minimal elements in a poset. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001729The number of visible descents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000021The number of descents of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000659The number of rises of length at least 2 of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001642The Prague dimension of a graph. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001812The biclique partition number of a graph. St001928The number of non-overlapping descents in a permutation. St000023The number of inner peaks of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000353The number of inner valleys of a permutation. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000658The number of rises of length 2 of a Dyck path. St000822The Hadwiger number of the graph. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001556The number of inversions of the third entry of a permutation. St001569The maximal modular displacement of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000382The first part of an integer composition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000908The length of the shortest maximal antichain in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St001472The permanent of the Coxeter matrix of the poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001368The number of vertices of maximal degree in a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001429The number of negative entries in a signed permutation. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000629The defect of a binary word. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001895The oddness of a signed permutation. St000022The number of fixed points of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000534The number of 2-rises of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000842The breadth of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St000878The number of ones minus the number of zeros of a binary word. St001867The number of alignments of type EN of a signed permutation. St000655The length of the minimal rise of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000007The number of saliances of the permutation. St000297The number of leading ones in a binary word. St000366The number of double descents of a permutation. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000665The number of rafts of a permutation. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000982The length of the longest constant subword. St001115The number of even descents of a permutation. St001310The number of induced diamond graphs in a graph. St001644The dimension of a graph. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000047The number of standard immaculate tableaux of a given shape. St000056The decomposition (or block) number of a permutation. St000277The number of ribbon shaped standard tableaux. St000287The number of connected components of a graph. St000295The length of the border of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000381The largest part of an integer composition. St000383The last part of an integer composition. St000409The number of pitchforks in a binary tree. St000570The Edelman-Greene number of a permutation. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000701The protection number of a binary tree. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000808The number of up steps of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000820The number of compositions obtained by rotating the composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001256Number of simple reflexive modules that are 2-stable reflexive. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001765The number of connected components of the friends and strangers graph. St001768The number of reduced words of a signed permutation. St001820The size of the image of the pop stack sorting operator. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St000145The Dyson rank of a partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000221The number of strong fixed points of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000355The number of occurrences of the pattern 21-3. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000650The number of 3-rises of a permutation. St000666The number of right tethers of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000783The side length of the largest staircase partition fitting into a partition. St000787The number of flips required to make a perfect matching noncrossing. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000894The trace of an alternating sign matrix. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001172The number of 1-rises at odd height of a Dyck path. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001486The number of corners of the ribbon associated with an integer composition. St001520The number of strict 3-descents. St001524The degree of symmetry of a binary word. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001584The area statistic between a Dyck path and its bounce path. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001673The degree of asymmetry of an integer composition. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001703The villainy of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001846The number of elements which do not have a complement in the lattice. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001871The number of triconnected components of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition.