- FindStat

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000203
(load all 4 compositions to match this statistic)

Mapping

St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of external nodes of a binary tree. That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus

$1$ for the root node itself. A counting formula for the number of external node in all binary trees of size

$n$ can be found in [1].

Matching statistic: St000056

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [2,1] => [2,1] => [1] => 1
[[.,.],.]
 => [1,2] => [1,2] => [1] => 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [1,2] => 2
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [2,1] => 1
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,2] => 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1] => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2] => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [1,2,3] => 3
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [1,3,2] => 2
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [2,1,3] => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [3,2,1] => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [2,3,1] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [1,2,3] => 3
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [1,3,2] => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,2,3] => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [3,1,2] => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1] => 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2] => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 3
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 4
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 3
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 3
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 3
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 2
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 4
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 3
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 3
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 2
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 3
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 4
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 3
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 2
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 2
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 3
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 3
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1

Description

The decomposition (or block) number of a permutation. For

$\pi \in \mathcal{S}_n$ , this is given by

$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].

Matching statistic: St000234

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of global ascents of a permutation. The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$ Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$ For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$ or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].

Matching statistic: St001879
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

Matching statistic: St001880
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.