- FindStat

Your data matches 481 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001719
(load all 12 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval

$[a, b]$ in a lattice is small if

$b$ is a join of elements covering

$a$ .

Matching statistic: St001095
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001095: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-isomorphic posets with precisely one further covering relation.

Matching statistic: St001942
(load all 2 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001942: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of loops of the quiver corresponding to the reduced incidence algebra of a poset.

Matching statistic: St001964
(load all 5 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 91%●distinct values known / distinct values provided: 50%

Values

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

Description

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

Matching statistic: St000181
(load all 73 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 72%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of connected components of the Hasse diagram for the poset.

Matching statistic: St001890
(load all 113 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 72%●distinct values known / distinct values provided: 50%

Values

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

Description

The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value

$\mu(x, y)$ is equal to the signed sum of chains from

$x$ to

$y$ , where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.

Matching statistic: St001933

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => [1]
 => 1 = 0 + 1
[[.,.],.]
 => ([(0,1)],2)
 => [1]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [1]
 => 1 = 0 + 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [1]
 => 1 = 0 + 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [1]
 => 1 = 0 + 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [1]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0 + 1
[[[.,.],.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0 + 1
[[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0 + 1
[[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => [10,10]
 => ? = 0 + 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => [10,10]
 => ? = 0 + 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => [10,10]
 => ? = 0 + 1
[.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => [10,10]
 => ? = 0 + 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[.,.],[[.,.],[[[.,.],.],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0 + 1
[[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0 + 1
[[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0 + 1
[[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0 + 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[.,.],[[[[.,.],.],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[.,.],[[[.,.],[.,[.,.]]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[.,.],[[[.,.],[[.,.],.]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[.,.],[[[[.,.],.],[.,.]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[.,[.,.]],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => ? = 0 + 1
[[.,[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => ? = 0 + 1
[[.,[.,.]],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => ? = 0 + 1
[[.,[.,.]],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => ? = 0 + 1
[[.,[.,.]],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => ? = 0 + 1
[[.,[.,.]],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => ? = 0 + 1
[[[.,.],.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => ? = 0 + 1
[[[.,.],.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => ? = 0 + 1
[[[.,.],.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => ? = 0 + 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => ? = 0 + 1
[[[.,.],.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => ? = 0 + 1
[[[.,.],.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => ? = 0 + 1
[[.,[.,[.,.]]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => ? = 0 + 1
[[.,[[.,.],.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => ? = 0 + 1
[[[.,.],[.,.]],[.,[.,[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => ? = 0 + 1
[[[.,.],[.,.]],[.,[[.,.],.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => ? = 0 + 1
[[[.,.],[.,.]],[[.,[.,.]],.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => ? = 0 + 1
[[[.,.],[.,.]],[[[.,.],.],.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => ? = 0 + 1
[[[.,[.,.]],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => ? = 0 + 1
[[[[.,.],.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => ? = 0 + 1
[[.,[[.,.],[.,.]]],[.,[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => ? = 0 + 1
[[.,[[.,.],[.,.]]],[[.,.],.]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => ? = 0 + 1

Description

The largest multiplicity of a part in an integer partition.

Matching statistic: St001392

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => [1]
 => [1]
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => [1]
 => [1]
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 0
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 0
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 0
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [2,2,1,1]
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [2,2,1,1]
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [2,2,1,1]
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [2,2,1,1]
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0
[[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[[.,.],.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0
[.,[[[.,.],.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0
[.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0
[.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[.,.],[.,[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[.,.],[.,[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[.,.],[[[.,.],.],.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => [2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => [2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => [2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => [2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[[[.,.],.],.],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[[.,.],[.,[.,.]]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[[.,.],[[.,.],.]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[[.,[.,.]],[.,.]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,.],[[[[.,.],.],[.,.]],.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[[.,[.,.]],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => [6,6,6,4,4,4]
 => ? = 0
[[.,[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => [3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[.,[.,.]],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => [3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[.,[.,.]],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => [3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[.,[.,.]],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => [3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[.,[.,.]],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => [6,6,6,4,4,4]
 => ? = 0
[[[.,.],.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => [6,6,6,4,4,4]
 => ? = 0
[[[.,.],.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => [3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[[.,.],.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => [3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[[.,.],.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => [3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[[.,.],.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => [18,18,9]
 => [3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2]
 => ? = 0
[[[.,.],.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => [6,6,6,4,4,4]
 => ? = 0
[[.,[.,[.,.]]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => [4,4,4,4,3,3,3,3,3,3,3,3]
 => ? = 0
[[.,[[.,.],.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => [4,4,4,4,3,3,3,3,3,3,3,3]
 => ? = 0
[[[.,.],[.,.]],[.,[.,[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => [4,4,4,4,3,3,3,3,3,3,3,3]
 => ? = 0
[[[.,.],[.,.]],[.,[[.,.],.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => [4,4,4,4,3,3,3,3,3,3,3,3]
 => ? = 0
[[[.,.],[.,.]],[[.,[.,.]],.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => [4,4,4,4,3,3,3,3,3,3,3,3]
 => ? = 0
[[[.,.],[.,.]],[[[.,.],.],.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => [4,4,4,4,3,3,3,3,3,3,3,3]
 => ? = 0
[[[.,[.,.]],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => [4,4,4,4,3,3,3,3,3,3,3,3]
 => ? = 0
[[[[.,.],.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => [12,12,12,4]
 => [4,4,4,4,3,3,3,3,3,3,3,3]
 => ? = 0
[[.,[[.,.],[.,.]]],[.,[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => [6,6,6,4,4,4]
 => ? = 0
[[.,[[.,.],[.,.]]],[[.,.],.]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => [6,6,6,6,3,3]
 => [6,6,6,4,4,4]
 => ? = 0

Description

The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.

Matching statistic: St001301
(load all 20 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%

Values

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

Description

The first Betti number of the order complex associated with the poset. The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.

Matching statistic: St001396
(load all 6 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001396: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%

Values

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

Description

Number of triples of incomparable elements in a finite poset. For a finite poset this is the number of 3-element sets

$S \in \binom{P}{3}$ that are pairwise incomparable.

The following 471 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001091The number of parts in an integer partition whose next smaller part has the same size. St000160The multiplicity of the smallest part of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000478Another weight of a partition according to Alladi. St000759The smallest missing part in an integer partition. St001626The number of maximal proper sublattices of a lattice. St000475The number of parts equal to 1 in a partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001272The number of graphs with the same degree sequence. St001568The smallest positive integer that does not appear twice in the partition. St001271The competition number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001434The number of negative sum pairs of a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001260The permanent of an alternating sign matrix. St001116The game chromatic number of a graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St000322The skewness of a graph. St000068The number of minimal elements in a poset. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000623The number of occurrences of the pattern 52341 in a permutation. St001381The fertility of a permutation. St000546The number of global descents of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000054The first entry of the permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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. St001545The second Elser number of a connected graph. St000644The number of graphs with given frequency partition. 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. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St000002The number of occurrences of the pattern 123 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001625The Möbius invariant of a lattice. 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. St001394The genus of a permutation. St001621The number of atoms of a lattice. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000632The jump number of the poset. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. 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. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000666The number of right tethers of a permutation. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000363The number of minimal vertex covers of a graph. St000544The cop number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St001342The number of vertices in the center of a graph. St001363The Euler characteristic of a graph according to Knill. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001367The smallest number which does not occur as degree of a vertex in a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000210Minimum over maximum difference of elements in cycles. St000379The number of Hamiltonian cycles in a graph. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000456The monochromatic index of a connected graph. St000487The length of the shortest cycle of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001811The Castelnuovo-Mumford regularity of a 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

$K[x]/(x^n)$ . St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000221The number of strong fixed points of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings 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. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000450The number of edges minus the number of vertices plus 2 of a graph. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. 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. 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. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001810The number of fixed points of a permutation smaller than its largest moved point. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001871The number of triconnected components of a graph. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000694The number of affine bounded permutations that project to a given permutation. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. 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. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St001827The number of two-component spanning forests of a graph. St001828The Euler characteristic of a graph. 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. St000822The Hadwiger number of the graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000241The number of cyclical small excedances. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000317The cycle descent number of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000531The leading coefficient of the rook polynomial of an integer partition. St000552The number of cut vertices of a graph. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000664The number of right ropes of a permutation. St000674The number of hills 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. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St000989The number of final rises of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in 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. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001479The number of bridges of a graph. St001520The number of strict 3-descents. St001537The number of cyclic crossings 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. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001570The minimal number of edges to add to make a graph Hamiltonian. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001703The villainy of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001783The number of odd automorphisms of a graph. St001826The maximal number of leaves on a vertex of a graph. St001856The number of edges in the reduced word graph of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St001957The number of Hasse diagrams with a given underlying undirected graph. St000061The number of nodes on the left branch of a binary tree. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000273The domination number of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000553The number of blocks of a graph. St000570The Edelman-Greene number of a permutation. St000627The exponent of a binary word. St000654The first descent of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000776The maximal multiplicity of an eigenvalue in a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000843The decomposition number of a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000916The packing number of a graph. St000917The open packing number of a graph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St000991The number of right-to-left minima of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001081The number of minimal length factorizations of a permutation into star transpositions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001589The nesting number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001672The restrained domination number of a graph. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000062The length of the longest increasing subsequence of the permutation. St000258The burning number of a graph. St000272The treewidth of a graph. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000536The pathwidth of a graph. St000542The number of left-to-right-minima of a permutation. St000733The row containing the largest entry of a standard tableau. St000897The number of different multiplicities of parts of an integer partition. St000918The 2-limited packing number of a graph. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001111The weak 2-dynamic chromatic number of a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001277The degeneracy of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001358The largest degree of a regular subgraph of a graph. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001792The arboricity of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001846The number of elements which do not have a complement in the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001820The size of the image of the pop stack sorting operator. 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

$eA$ . St001720The minimal length of a chain of small intervals in a lattice. St000741The Colin de Verdière graph invariant. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000382The first part of an integer composition. St001866The nesting alignments of a signed permutation. 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. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001889The size of the connectivity set of a signed permutation. St001902The number of potential covers of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. 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. St000640The rank of the largest boolean interval 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. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001616The number of neutral elements in a lattice. 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. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001644The dimension of a graph. St001281The normalized isoperimetric number of a graph. St001118The acyclic chromatic index of a graph. St000383The last part of an integer composition. St000297The number of leading ones in a binary word. St000366The number of double descents of a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001115The number of even descents of a 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. St000439The position of the first down step of a Dyck path.