Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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
[[.,.],.]
 => [1,2] => [1,2] => [1] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [1,2] => 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [2,1] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,2] => 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2] => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [1,2,3] => 2
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [1,3,2] => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [2,1,3] => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [3,2,1] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [2,3,1] => 0
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [1,2,3] => 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [1,3,2] => 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,2,3] => 2
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [3,1,2] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 3
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 2
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 3
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 2
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 2
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 3
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3] => 0
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: St000203
(load all 4 compositions to match this statistic)
Mapping
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [[.,.],.]
 => 2 = 0 + 2
[[.,.],.]
 => [.,[.,.]]
 => 2 = 0 + 2
[.,[.,[.,.]]]
 => [[[.,.],.],.]
 => 3 = 1 + 2
[.,[[.,.],.]]
 => [[.,[.,.]],.]
 => 2 = 0 + 2
[[.,.],[.,.]]
 => [[.,.],[.,.]]
 => 3 = 1 + 2
[[.,[.,.]],.]
 => [.,[[.,.],.]]
 => 2 = 0 + 2
[[[.,.],.],.]
 => [.,[.,[.,.]]]
 => 3 = 1 + 2
[.,[.,[.,[.,.]]]]
 => [[[[.,.],.],.],.]
 => 4 = 2 + 2
[.,[.,[[.,.],.]]]
 => [[[.,[.,.]],.],.]
 => 3 = 1 + 2
[.,[[.,.],[.,.]]]
 => [[[.,.],[.,.]],.]
 => 3 = 1 + 2
[.,[[.,[.,.]],.]]
 => [[.,[[.,.],.]],.]
 => 2 = 0 + 2
[.,[[[.,.],.],.]]
 => [[.,[.,[.,.]]],.]
 => 2 = 0 + 2
[[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => 4 = 2 + 2
[[.,.],[[.,.],.]]
 => [[.,[.,.]],[.,.]]
 => 3 = 1 + 2
[[.,[.,.]],[.,.]]
 => [[.,.],[[.,.],.]]
 => 3 = 1 + 2
[[[.,.],.],[.,.]]
 => [[.,.],[.,[.,.]]]
 => 4 = 2 + 2
[[.,[.,[.,.]]],.]
 => [.,[[[.,.],.],.]]
 => 2 = 0 + 2
[[.,[[.,.],.]],.]
 => [.,[[.,[.,.]],.]]
 => 2 = 0 + 2
[[[.,.],[.,.]],.]
 => [.,[[.,.],[.,.]]]
 => 3 = 1 + 2
[[[.,[.,.]],.],.]
 => [.,[.,[[.,.],.]]]
 => 3 = 1 + 2
[[[[.,.],.],.],.]
 => [.,[.,[.,[.,.]]]]
 => 4 = 2 + 2
[.,[.,[.,[.,[.,.]]]]]
 => [[[[[.,.],.],.],.],.]
 => 5 = 3 + 2
[.,[.,[.,[[.,.],.]]]]
 => [[[[.,[.,.]],.],.],.]
 => 4 = 2 + 2
[.,[.,[[.,.],[.,.]]]]
 => [[[[.,.],[.,.]],.],.]
 => 4 = 2 + 2
[.,[.,[[.,[.,.]],.]]]
 => [[[.,[[.,.],.]],.],.]
 => 3 = 1 + 2
[.,[.,[[[.,.],.],.]]]
 => [[[.,[.,[.,.]]],.],.]
 => 3 = 1 + 2
[.,[[.,.],[.,[.,.]]]]
 => [[[[.,.],.],[.,.]],.]
 => 4 = 2 + 2
[.,[[.,.],[[.,.],.]]]
 => [[[.,[.,.]],[.,.]],.]
 => 3 = 1 + 2
[.,[[.,[.,.]],[.,.]]]
 => [[[.,.],[[.,.],.]],.]
 => 3 = 1 + 2
[.,[[[.,.],.],[.,.]]]
 => [[[.,.],[.,[.,.]]],.]
 => 3 = 1 + 2
[.,[[.,[.,[.,.]]],.]]
 => [[.,[[[.,.],.],.]],.]
 => 2 = 0 + 2
[.,[[.,[[.,.],.]],.]]
 => [[.,[[.,[.,.]],.]],.]
 => 2 = 0 + 2
[.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => 2 = 0 + 2
[.,[[[.,[.,.]],.],.]]
 => [[.,[.,[[.,.],.]]],.]
 => 2 = 0 + 2
[.,[[[[.,.],.],.],.]]
 => [[.,[.,[.,[.,.]]]],.]
 => 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
 => [[[[.,.],.],.],[.,.]]
 => 5 = 3 + 2
[[.,.],[.,[[.,.],.]]]
 => [[[.,[.,.]],.],[.,.]]
 => 4 = 2 + 2
[[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => 4 = 2 + 2
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],.]],[.,.]]
 => 3 = 1 + 2
[[.,.],[[[.,.],.],.]]
 => [[.,[.,[.,.]]],[.,.]]
 => 3 = 1 + 2
[[.,[.,.]],[.,[.,.]]]
 => [[[.,.],.],[[.,.],.]]
 => 4 = 2 + 2
[[.,[.,.]],[[.,.],.]]
 => [[.,[.,.]],[[.,.],.]]
 => 3 = 1 + 2
[[[.,.],.],[.,[.,.]]]
 => [[[.,.],.],[.,[.,.]]]
 => 5 = 3 + 2
[[[.,.],.],[[.,.],.]]
 => [[.,[.,.]],[.,[.,.]]]
 => 4 = 2 + 2
[[.,[.,[.,.]]],[.,.]]
 => [[.,.],[[[.,.],.],.]]
 => 3 = 1 + 2
[[.,[[.,.],.]],[.,.]]
 => [[.,.],[[.,[.,.]],.]]
 => 3 = 1 + 2
[[[.,.],[.,.]],[.,.]]
 => [[.,.],[[.,.],[.,.]]]
 => 4 = 2 + 2
[[[.,[.,.]],.],[.,.]]
 => [[.,.],[.,[[.,.],.]]]
 => 4 = 2 + 2
[[[[.,.],.],.],[.,.]]
 => [[.,.],[.,[.,[.,.]]]]
 => 5 = 3 + 2
[[.,[.,[.,[.,.]]]],.]
 => [.,[[[[.,.],.],.],.]]
 => 2 = 0 + 2
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 5 + 2
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
 => [[[[[[.,[[.,.],.]],.],.],.],.],.]
 => ? = 4 + 2
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
 => [[[[[[.,.],[[.,.],.]],.],.],.],.]
 => ? = 4 + 2
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
 => [[[[[.,[[[.,.],.],.]],.],.],.],.]
 => ? = 3 + 2
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
 => [[[[[.,[[.,[.,.]],.]],.],.],.],.]
 => ? = 3 + 2
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
 => [[[[[.,[[.,.],[.,.]]],.],.],.],.]
 => ? = 3 + 2
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
 => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
 => ? = 3 + 2
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
 => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
 => ? = 3 + 2
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
 => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
 => ? = 4 + 2
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
 => [[[[[.,[[.,.],.]],[.,.]],.],.],.]
 => ? = 3 + 2
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
 => [[[[[.,.],[[.,[.,.]],.]],.],.],.]
 => ? = 3 + 2
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
 => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
 => ? = 3 + 2
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
 => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
 => ? = 4 + 2
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
 => [[[[[.,[[.,.],.]],.],[.,.]],.],.]
 => ? = 3 + 2
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
 => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
 => ? = 3 + 2
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
 => [[[[[.,.],[[.,.],.]],[.,.]],.],.]
 => ? = 3 + 2
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
 => [[[[[.,[.,.]],.],[[.,.],.]],.],.]
 => ? = 3 + 2
[.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
 => [[[[[.,.],[.,.]],[[.,.],.]],.],.]
 => ? = 3 + 2
[.,[.,[[.,[.,.]],[[.,[.,.]],.]]]]
 => [[[[.,[[.,.],.]],[[.,.],.]],.],.]
 => ? = 2 + 2
[.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
 => [[[[.,[[.,.],.]],[.,[.,.]]],.],.]
 => ? = 2 + 2
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
 => [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
 => ? = 2 + 2
[.,[.,[[.,[.,[.,.]]],[[.,.],.]]]]
 => [[[[.,[.,.]],[[[.,.],.],.]],.],.]
 => ? = 2 + 2
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
 => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
 => ? = 1 + 2
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
 => [[[[[[[.,.],.],.],.],.],[.,.]],.]
 => ? = 5 + 2
[.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
 => [[[[[.,.],[[.,.],.]],.],[.,.]],.]
 => ? = 3 + 2
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
 => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
 => ? = 3 + 2
[.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
 => [[[[.,[.,.]],[[.,.],.]],[.,.]],.]
 => ? = 2 + 2
[.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]
 => [[[[[.,.],[.,.]],.],[[.,.],.]],.]
 => ? = 3 + 2
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
 => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
 => ? = 4 + 2
[.,[[[.,.],.],[[.,.],[[.,.],.]]]]
 => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
 => ? = 2 + 2
[.,[[[.,.],.],[[.,[.,[.,.]]],.]]]
 => [[[.,[[[.,.],.],.]],[.,[.,.]]],.]
 => ? = 1 + 2
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
 => [[[[[.,.],.],.],[[.,[.,.]],.]],.]
 => ? = 3 + 2
[.,[[.,[[.,.],.]],[.,[[.,.],.]]]]
 => [[[[.,[.,.]],.],[[.,[.,.]],.]],.]
 => ? = 2 + 2
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
 => [[[[[.,.],.],.],[.,[.,[.,.]]]],.]
 => ? = 3 + 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
 => [[[.,.],[[.,[[.,[.,.]],.]],.]],.]
 => ? = 1 + 2
[.,[[.,[[.,[[[.,.],.],.]],.]],.]]
 => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
 => ? = 0 + 2
[.,[[.,[[[.,[[.,.],.]],.],.]],.]]
 => [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
 => ? = 0 + 2
[.,[[.,[[[[.,.],[.,.]],.],.]],.]]
 => [[.,[[.,[.,[[.,.],[.,.]]]],.]],.]
 => ? = 0 + 2
[.,[[[.,.],[[[.,.],[.,.]],.]],.]]
 => [[.,[[.,[[.,.],[.,.]]],[.,.]]],.]
 => ? = 0 + 2
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
 => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => ? = 4 + 2
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
 => [[.,[[[[[.,.],.],.],.],.]],[.,.]]
 => ? = 1 + 2
[[.,.],[[.,[[.,[.,.]],[.,.]]],.]]
 => [[.,[[[.,.],[[.,.],.]],.]],[.,.]]
 => ? = 1 + 2
[[.,.],[[.,[[[.,.],.],[.,.]]],.]]
 => [[.,[[[.,.],[.,[.,.]]],.]],[.,.]]
 => ? = 1 + 2
[[.,.],[[.,[[[.,[.,.]],.],.]],.]]
 => [[.,[[.,[.,[[.,.],.]]],.]],[.,.]]
 => ? = 1 + 2
[[.,.],[[.,[[[[.,.],.],.],.]],.]]
 => [[.,[[.,[.,[.,[.,.]]]],.]],[.,.]]
 => ? = 1 + 2
[[.,.],[[[[.,[[.,.],.]],.],.],.]]
 => [[.,[.,[.,[[.,[.,.]],.]]]],[.,.]]
 => ? = 1 + 2
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
 => [[.,[.,[.,[.,[[.,.],.]]]]],[.,.]]
 => ? = 1 + 2
[[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => [[[[.,[.,[.,.]]],.],.],[[.,.],.]]
 => ? = 3 + 2
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
 => [[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
 => ? = 2 + 2
[[.,[.,.]],[[[.,.],[.,[.,.]]],.]]
 => [[.,[[[.,.],.],[.,.]]],[[.,.],.]]
 => ? = 1 + 2
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: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 86%
Values
[.,[.,.]]
 => [2,1] => [2,1] => [1] => 1 = 0 + 1
[[.,.],.]
 => [1,2] => [1,2] => [1] => 1 = 0 + 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [1,2] => 2 = 1 + 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [2,1] => 1 = 0 + 1
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,2] => 2 = 1 + 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1] => 1 = 0 + 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2] => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [1,3,2] => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [2,1,3] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [2,3,1] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [1,2,3] => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [1,3,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [3,1,2] => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1] => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 4 = 3 + 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [8,7,6,5,4,3,2,1] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
 => [7,8,6,5,4,3,2,1] => [8,1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 5 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
 => [7,6,8,5,4,3,2,1] => [8,1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 4 + 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
 => [6,5,8,7,4,3,2,1] => [8,1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 4 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
 => [7,6,5,8,4,3,2,1] => [8,1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 3 + 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
 => [6,7,5,8,4,3,2,1] => [8,1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
 => [5,7,6,8,4,3,2,1] => [8,1,2,3,5,7,6,4] => [1,2,3,5,7,6,4] => ? = 3 + 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
 => [6,5,7,8,4,3,2,1] => [8,1,2,3,6,5,7,4] => [1,2,3,6,5,7,4] => ? = 3 + 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
 => [5,6,7,8,4,3,2,1] => [8,1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => ? = 3 + 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
 => [4,6,8,7,5,3,2,1] => [8,1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 4 + 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
 => [4,7,6,8,5,3,2,1] => [8,1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 3 + 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
 => [5,6,4,8,7,3,2,1] => [8,1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 3 + 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
 => [4,5,6,8,7,3,2,1] => [8,1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => ? = 3 + 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
 => [3,7,8,6,5,4,2,1] => [8,1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ? = 4 + 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
 => [3,7,6,8,5,4,2,1] => [8,1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => ? = 3 + 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
 => [3,5,8,7,6,4,2,1] => [8,1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 4 + 1
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
 => [3,5,7,8,6,4,2,1] => [8,1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => ? = 3 + 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
 => [3,6,5,8,7,4,2,1] => [8,1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => ? = 3 + 1
[.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
 => [4,3,8,7,6,5,2,1] => [8,1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => ? = 4 + 1
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
 => [4,3,7,8,6,5,2,1] => [8,1,4,3,2,5,7,6] => [1,4,3,2,5,7,6] => ? = 3 + 1
[.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
 => [4,3,6,8,7,5,2,1] => [8,1,4,3,2,6,5,7] => [1,4,3,2,6,5,7] => ? = 3 + 1
[.,[.,[[.,[.,.]],[[.,[.,.]],.]]]]
 => [4,3,7,6,8,5,2,1] => [8,1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => ? = 2 + 1
[.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
 => [3,4,7,6,8,5,2,1] => [8,1,3,4,2,7,6,5] => [1,3,4,2,7,6,5] => ? = 2 + 1
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
 => [3,4,6,7,8,5,2,1] => [8,1,3,4,2,6,7,5] => [1,3,4,2,6,7,5] => ? = 2 + 1
[.,[.,[[.,[.,[.,.]]],[[.,.],.]]]]
 => [5,4,3,7,8,6,2,1] => [8,1,5,3,4,2,7,6] => [1,5,3,4,2,7,6] => ? = 2 + 1
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
 => [3,4,5,6,7,8,2,1] => [8,1,3,4,5,6,7,2] => [1,3,4,5,6,7,2] => ? = 1 + 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
 => [2,8,7,6,5,4,3,1] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 5 + 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
 => [2,6,8,7,5,4,3,1] => [8,2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 4 + 1
[.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
 => [2,6,5,8,7,4,3,1] => [8,2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => ? = 3 + 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
 => [2,4,6,8,7,5,3,1] => [8,2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3 + 1
[.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
 => [2,5,4,7,8,6,3,1] => [8,2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 2 + 1
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
 => [3,2,8,7,6,5,4,1] => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 4 + 1
[.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]
 => [3,2,6,8,7,5,4,1] => [8,3,2,1,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 3 + 1
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
 => [3,2,6,5,8,7,4,1] => [8,3,2,1,6,5,4,7] => [3,2,1,6,5,4,7] => ? = 2 + 1
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
 => [2,3,8,7,6,5,4,1] => [8,2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 4 + 1
[.,[[[.,.],.],[[.,.],[[.,.],.]]]]
 => [2,3,5,7,8,6,4,1] => [8,2,3,1,5,4,7,6] => [2,3,1,5,4,7,6] => ? = 2 + 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
 => [2,3,5,6,8,7,4,1] => [8,2,3,1,5,6,4,7] => [2,3,1,5,6,4,7] => ? = 2 + 1
[.,[[[.,.],.],[[.,[.,[.,.]]],.]]]
 => [2,3,7,6,5,8,4,1] => [8,2,3,1,7,5,6,4] => [2,3,1,7,5,6,4] => ? = 1 + 1
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
 => [3,4,2,8,7,6,5,1] => [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 3 + 1
[.,[[.,[[.,.],.]],[.,[[.,.],.]]]]
 => [3,4,2,7,8,6,5,1] => [8,4,3,2,1,5,7,6] => [4,3,2,1,5,7,6] => ? = 2 + 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
 => [2,3,4,8,7,6,5,1] => [8,2,3,4,1,5,6,7] => [2,3,4,1,5,6,7] => ? = 3 + 1
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
 => [4,5,3,6,2,8,7,1] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 1 + 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [5,4,6,3,7,2,8,1] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[[.,[[.,[[[.,.],.],.]],.]],.]]
 => [4,5,6,3,7,2,8,1] => [8,7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => ? = 0 + 1
[.,[[.,[[[.,.],[[.,.],.]],.]],.]]
 => [3,5,6,4,7,2,8,1] => [8,7,3,6,5,4,2,1] => [7,3,6,5,4,2,1] => ? = 0 + 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [4,3,6,5,7,2,8,1] => [8,7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => ? = 0 + 1
[.,[[.,[[[.,[[.,.],.]],.],.]],.]]
 => [4,5,3,6,7,2,8,1] => [8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => ? = 0 + 1
[.,[[.,[[[[.,.],[.,.]],.],.]],.]]
 => [3,5,4,6,7,2,8,1] => [8,7,3,5,4,6,2,1] => [7,3,5,4,6,2,1] => ? = 0 + 1
[.,[[[.,.],[[[.,.],[.,.]],.]],.]]
 => [2,4,6,5,7,3,8,1] => [8,2,7,4,6,5,3,1] => [2,7,4,6,5,3,1] => ? = 0 + 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
 => [3,2,6,5,7,4,8,1] => [8,3,2,7,6,5,4,1] => [3,2,7,6,5,4,1] => ? = 0 + 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: 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: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 71%
Values
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ? = 0 + 1
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 0 + 1
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[[.,.],[.,.]]
 => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 1
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 1
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 1
[[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 1
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 1
[[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 1
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 1
[[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 1
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[.,.],.],[.,[.,[.,.]]]]
 => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[[.,.],.],.],[.,[.,.]]]
 => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[[[.,.],.],.],.],[.,.]]
 => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[.,.],.],.],[.,[.,[.,.]]]]
 => [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[[[[.,.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: 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: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 71%
Values
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ? = 0 + 2
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 0 + 2
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[[.,.],[.,.]]
 => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 0 + 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 2
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 2
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 0 + 2
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 1 + 2
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 2
[[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 2
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[.,.],.],[.,[.,[.,.]]]]
 => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[[.,.],.],.],[.,[.,.]]]
 => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[[[.,.],.],.],.],[.,.]]
 => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[.,.],.],.],[.,[.,[.,.]]]]
 => [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[[[[.,.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.