Your data matches 5 different statistics following compositions of up to 3 maps.
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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
[.,.]
 => 1
[.,[.,.]]
 => 2
[[.,.],.]
 => 2
[.,[.,[.,.]]]
 => 3
[.,[[.,.],.]]
 => 2
[[.,.],[.,.]]
 => 3
[[.,[.,.]],.]
 => 2
[[[.,.],.],.]
 => 3
[.,[.,[.,[.,.]]]]
 => 4
[.,[.,[[.,.],.]]]
 => 3
[.,[[.,.],[.,.]]]
 => 3
[.,[[.,[.,.]],.]]
 => 2
[.,[[[.,.],.],.]]
 => 2
[[.,.],[.,[.,.]]]
 => 4
[[.,.],[[.,.],.]]
 => 3
[[.,[.,.]],[.,.]]
 => 3
[[[.,.],.],[.,.]]
 => 4
[[.,[.,[.,.]]],.]
 => 2
[[.,[[.,.],.]],.]
 => 2
[[[.,.],[.,.]],.]
 => 3
[[[.,[.,.]],.],.]
 => 3
[[[[.,.],.],.],.]
 => 4
[.,[.,[.,[.,[.,.]]]]]
 => 5
[.,[.,[.,[[.,.],.]]]]
 => 4
[.,[.,[[.,.],[.,.]]]]
 => 4
[.,[.,[[.,[.,.]],.]]]
 => 3
[.,[.,[[[.,.],.],.]]]
 => 3
[.,[[.,.],[.,[.,.]]]]
 => 4
[.,[[.,.],[[.,.],.]]]
 => 3
[.,[[.,[.,.]],[.,.]]]
 => 3
[.,[[[.,.],.],[.,.]]]
 => 3
[.,[[.,[.,[.,.]]],.]]
 => 2
[.,[[.,[[.,.],.]],.]]
 => 2
[.,[[[.,.],[.,.]],.]]
 => 2
[.,[[[.,[.,.]],.],.]]
 => 2
[.,[[[[.,.],.],.],.]]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => 5
[[.,.],[.,[[.,.],.]]]
 => 4
[[.,.],[[.,.],[.,.]]]
 => 4
[[.,.],[[.,[.,.]],.]]
 => 3
[[.,.],[[[.,.],.],.]]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => 4
[[.,[.,.]],[[.,.],.]]
 => 3
[[[.,.],.],[.,[.,.]]]
 => 5
[[[.,.],.],[[.,.],.]]
 => 4
[[.,[.,[.,.]]],[.,.]]
 => 3
[[.,[[.,.],.]],[.,.]]
 => 3
[[[.,.],[.,.]],[.,.]]
 => 4
[[[.,[.,.]],.],[.,.]]
 => 4
[[[[.,.],.],.],[.,.]]
 => 5
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: St000234
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Values
[.,.]
 => [1] => [1] => [] => ? = 1 - 2
[.,[.,.]]
 => [2,1] => [2,1] => [1] => 0 = 2 - 2
[[.,.],.]
 => [1,2] => [1,2] => [1] => 0 = 2 - 2
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [1,2] => 1 = 3 - 2
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [2,1] => 0 = 2 - 2
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,2] => 1 = 3 - 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1] => 0 = 2 - 2
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2] => 1 = 3 - 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [1,2,3] => 2 = 4 - 2
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [1,3,2] => 1 = 3 - 2
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [2,1,3] => 1 = 3 - 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [3,2,1] => 0 = 2 - 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [2,3,1] => 0 = 2 - 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [1,2,3] => 2 = 4 - 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [1,3,2] => 1 = 3 - 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1 = 3 - 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,2,3] => 2 = 4 - 2
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [3,1,2] => 0 = 2 - 2
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1] => 0 = 2 - 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1 = 3 - 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1 = 3 - 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 2 = 4 - 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 1 = 3 - 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 1 = 3 - 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 2 = 4 - 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 1 = 3 - 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 1 = 3 - 2
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 1 = 3 - 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 0 = 2 - 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 0 = 2 - 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 0 = 2 - 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 0 = 2 - 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 0 = 2 - 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 1 = 3 - 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 1 = 3 - 2
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 2 = 4 - 2
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 1 = 3 - 2
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 3 = 5 - 2
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 2 = 4 - 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 1 = 3 - 2
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 1 = 3 - 2
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 2 = 4 - 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 2 = 4 - 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 3 = 5 - 2
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3] => 0 = 2 - 2
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
 => [2,4,3,8,7,6,5,1] => [8,2,4,3,1,5,6,7] => ? => ? = 5 - 2
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
 => [2,4,3,7,8,6,5,1] => [8,2,4,3,1,5,7,6] => ? => ? = 4 - 2
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
 => [2,4,3,6,8,7,5,1] => [8,2,4,3,1,6,5,7] => ? => ? = 4 - 2
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
 => [2,4,3,7,6,8,5,1] => [8,2,4,3,1,7,6,5] => ? => ? = 3 - 2
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
 => [2,4,3,6,7,8,5,1] => [8,2,4,3,1,6,7,5] => ? => ? = 3 - 2
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
 => [2,3,5,4,7,8,6,1] => [8,2,3,5,4,1,7,6] => ? => ? = 3 - 2
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
 => [2,4,3,5,7,8,6,1] => [8,2,4,3,5,1,7,6] => ? => ? = 3 - 2
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
 => [2,4,3,6,5,8,7,1] => [8,2,4,3,6,5,1,7] => ? => ? = 3 - 2
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
 => [3,5,4,2,6,8,7,1] => [8,5,3,2,4,6,1,7] => ? => ? = 3 - 2
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
 => [3,5,4,7,6,2,8,1] => [8,7,3,5,4,2,6,1] => ? => ? = 2 - 2
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
 => [4,6,5,3,7,2,8,1] => [8,7,6,4,3,5,2,1] => ? => ? = 2 - 2
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
 => [3,4,5,2,6,7,8,1] => [8,5,3,4,2,6,7,1] => ? => ? = 2 - 2
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
 => [2,3,5,4,6,7,8,1] => [8,2,3,5,4,6,7,1] => ? => ? = 2 - 2
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
 => [4,3,2,5,6,7,8,1] => [8,4,2,3,5,6,7,1] => ? => ? = 2 - 2
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
 => [3,4,2,5,6,7,8,1] => [8,4,3,2,5,6,7,1] => ? => ? = 2 - 2
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
 => [2,4,3,5,6,7,8,1] => [8,2,4,3,5,6,7,1] => ? => ? = 2 - 2
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
 => [3,2,4,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => ? => ? = 2 - 2
[.,[[[[[[[.,.],.],.],.],.],.],.]]
 => [2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => ? => ? = 2 - 2
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,7,6,8,5,4,3,2] => [1,8,2,3,4,7,6,5] => ? => ? = 6 - 2
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
 => [1,6,7,8,5,4,3,2] => [1,8,2,3,4,6,7,5] => ? => ? = 6 - 2
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,5,8,7,6,4,3,2] => [1,8,2,3,5,4,6,7] => ? => ? = 7 - 2
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,7,6,5,8,4,3,2] => [1,8,2,3,7,5,6,4] => ? => ? = 5 - 2
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,3,6,5,8,7,4,2] => [1,8,3,2,6,5,4,7] => ? => ? = 5 - 2
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
 => [1,3,5,6,7,8,4,2] => [1,8,3,2,5,6,7,4] => ? => ? = 4 - 2
[[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
 => [1,4,3,7,8,6,5,2] => [1,8,4,3,2,5,7,6] => ? => ? = 5 - 2
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,4,3,6,8,7,5,2] => [1,8,4,3,2,6,5,7] => ? => ? = 5 - 2
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
 => [1,4,3,6,7,8,5,2] => [1,8,4,3,2,6,7,5] => ? => ? = 4 - 2
[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
 => [1,5,4,3,8,7,6,2] => [1,8,5,3,4,2,6,7] => ? => ? = 5 - 2
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
 => [1,3,4,5,7,8,6,2] => [1,8,3,4,5,2,7,6] => ? => ? = 4 - 2
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
 => [1,3,4,6,5,8,7,2] => [1,8,3,4,6,5,2,7] => ? => ? = 4 - 2
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,3,5,4,6,8,7,2] => [1,8,3,5,4,6,2,7] => ? => ? = 4 - 2
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
 => [1,4,3,5,6,8,7,2] => [1,8,4,3,5,6,2,7] => ? => ? = 4 - 2
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,3,4,6,7,5,8,2] => [1,8,3,4,7,6,5,2] => ? => ? = 3 - 2
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
 => [1,3,5,4,7,6,8,2] => [1,8,3,5,4,7,6,2] => ? => ? = 3 - 2
[[.,.],[[[[[[.,.],.],.],.],.],.]]
 => [1,3,4,5,6,7,8,2] => [1,8,3,4,5,6,7,2] => ? => ? = 3 - 2
[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
 => [2,1,6,5,8,7,4,3] => [2,1,8,3,6,5,4,7] => ? => ? = 5 - 2
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
 => [2,1,5,6,7,8,4,3] => [2,1,8,3,5,6,7,4] => ? => ? = 4 - 2
[[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
 => [2,1,4,7,8,6,5,3] => [2,1,8,4,3,5,7,6] => ? => ? = 5 - 2
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
 => [2,1,4,6,7,8,5,3] => [2,1,8,4,3,6,7,5] => ? => ? = 4 - 2
[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
 => [2,1,5,4,8,7,6,3] => [2,1,8,5,4,3,6,7] => ? => ? = 5 - 2
[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
 => [2,1,4,6,5,8,7,3] => [2,1,8,4,6,5,3,7] => ? => ? = 4 - 2
[[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
 => [2,1,5,4,6,8,7,3] => [2,1,8,5,4,6,3,7] => ? => ? = 4 - 2
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
 => [2,1,4,5,6,8,7,3] => [2,1,8,4,5,6,3,7] => ? => ? = 4 - 2
[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]
 => [2,1,5,7,6,4,8,3] => [2,1,8,7,5,4,6,3] => ? => ? = 3 - 2
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
 => [2,1,4,6,7,5,8,3] => [2,1,8,4,7,6,5,3] => ? => ? = 3 - 2
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
 => [2,1,5,4,7,6,8,3] => [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,3] => ? = 3 - 2
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
 => [2,1,4,5,6,7,8,3] => [2,1,8,4,5,6,7,3] => ? => ? = 3 - 2
[[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
 => [3,2,1,7,8,6,5,4] => [3,1,2,8,4,5,7,6] => ? => ? = 5 - 2
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
 => [3,2,1,6,8,7,5,4] => [3,1,2,8,4,6,5,7] => ? => ? = 5 - 2
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: St000056
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 55%
Values
[.,.]
 => [1] => [1] => [] => ? = 1 - 1
[.,[.,.]]
 => [2,1] => [2,1] => [1] => 1 = 2 - 1
[[.,.],.]
 => [1,2] => [1,2] => [1] => 1 = 2 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [1,2] => 2 = 3 - 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [2,1] => 1 = 2 - 1
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,2] => 2 = 3 - 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1] => 1 = 2 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2] => 2 = 3 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [1,2,3] => 3 = 4 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [1,3,2] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [2,1,3] => 2 = 3 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [3,2,1] => 1 = 2 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [2,3,1] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [1,2,3] => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [1,3,2] => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,2,3] => 3 = 4 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1] => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2] => 2 = 3 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 3 = 4 - 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 3 = 4 - 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 4 = 5 - 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
 => [2,4,3,8,7,6,5,1] => [8,2,4,3,1,5,6,7] => ? => ? = 5 - 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
 => [2,4,3,7,8,6,5,1] => [8,2,4,3,1,5,7,6] => ? => ? = 4 - 1
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
 => [2,4,3,6,8,7,5,1] => [8,2,4,3,1,6,5,7] => ? => ? = 4 - 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
 => [2,4,3,7,6,8,5,1] => [8,2,4,3,1,7,6,5] => ? => ? = 3 - 1
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
 => [2,4,3,6,7,8,5,1] => [8,2,4,3,1,6,7,5] => ? => ? = 3 - 1
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
 => [2,3,5,4,7,8,6,1] => [8,2,3,5,4,1,7,6] => ? => ? = 3 - 1
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
 => [2,4,3,5,7,8,6,1] => [8,2,4,3,5,1,7,6] => ? => ? = 3 - 1
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
 => [2,4,3,6,5,8,7,1] => [8,2,4,3,6,5,1,7] => ? => ? = 3 - 1
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
 => [3,5,4,2,6,8,7,1] => [8,5,3,2,4,6,1,7] => ? => ? = 3 - 1
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
 => [3,5,4,7,6,2,8,1] => [8,7,3,5,4,2,6,1] => ? => ? = 2 - 1
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
 => [4,6,5,3,7,2,8,1] => [8,7,6,4,3,5,2,1] => ? => ? = 2 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [5,4,6,3,7,2,8,1] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 2 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [4,3,6,5,7,2,8,1] => [8,7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => ? = 2 - 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
 => [3,2,6,5,7,4,8,1] => [8,3,2,7,6,5,4,1] => [3,2,7,6,5,4,1] => ? = 2 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
 => [4,3,5,2,7,6,8,1] => [8,5,4,3,2,7,6,1] => [5,4,3,2,7,6,1] => ? = 2 - 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
 => [3,2,5,4,7,6,8,1] => [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,1] => ? = 2 - 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
 => [3,4,5,2,6,7,8,1] => [8,5,3,4,2,6,7,1] => ? => ? = 2 - 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
 => [2,3,5,4,6,7,8,1] => [8,2,3,5,4,6,7,1] => ? => ? = 2 - 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
 => [4,3,2,5,6,7,8,1] => [8,4,2,3,5,6,7,1] => ? => ? = 2 - 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
 => [3,4,2,5,6,7,8,1] => [8,4,3,2,5,6,7,1] => ? => ? = 2 - 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
 => [2,4,3,5,6,7,8,1] => [8,2,4,3,5,6,7,1] => ? => ? = 2 - 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
 => [3,2,4,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => ? => ? = 2 - 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
 => [2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => ? => ? = 2 - 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,8,7,6,5,4,3,2] => [1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 8 - 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,7,8,6,5,4,3,2] => [1,8,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 7 - 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,6,8,7,5,4,3,2] => [1,8,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 7 - 1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,7,6,8,5,4,3,2] => [1,8,2,3,4,7,6,5] => ? => ? = 6 - 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
 => [1,6,7,8,5,4,3,2] => [1,8,2,3,4,6,7,5] => ? => ? = 6 - 1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,5,8,7,6,4,3,2] => [1,8,2,3,5,4,6,7] => ? => ? = 7 - 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,7,6,5,8,4,3,2] => [1,8,2,3,7,5,6,4] => ? => ? = 5 - 1
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
 => [1,3,5,7,8,6,4,2] => [1,8,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => ? = 5 - 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,3,6,5,8,7,4,2] => [1,8,3,2,6,5,4,7] => ? => ? = 5 - 1
[[.,.],[[.,.],[[.,[[.,.],.]],.]]]
 => [1,3,6,7,5,8,4,2] => [1,8,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => ? = 4 - 1
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
 => [1,3,5,6,7,8,4,2] => [1,8,3,2,5,6,7,4] => ? => ? = 4 - 1
[[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
 => [1,4,3,7,8,6,5,2] => [1,8,4,3,2,5,7,6] => ? => ? = 5 - 1
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,4,3,6,8,7,5,2] => [1,8,4,3,2,6,5,7] => ? => ? = 5 - 1
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
 => [1,4,3,6,7,8,5,2] => [1,8,4,3,2,6,7,5] => ? => ? = 4 - 1
[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
 => [1,5,4,3,8,7,6,2] => [1,8,5,3,4,2,6,7] => ? => ? = 5 - 1
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
 => [1,4,5,3,7,8,6,2] => [1,8,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 4 - 1
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
 => [1,3,4,5,7,8,6,2] => [1,8,3,4,5,2,7,6] => ? => ? = 4 - 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
 => [1,3,4,6,5,8,7,2] => [1,8,3,4,6,5,2,7] => ? => ? = 4 - 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,3,5,4,6,8,7,2] => [1,8,3,5,4,6,2,7] => ? => ? = 4 - 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
 => [1,4,3,5,6,8,7,2] => [1,8,4,3,5,6,2,7] => ? => ? = 4 - 1
[[.,.],[[.,[[.,.],[[.,.],.]]],.]]
 => [1,4,6,7,5,3,8,2] => [1,8,7,4,3,6,5,2] => [1,7,4,3,6,5,2] => ? = 3 - 1
[[.,.],[[.,[[.,[[.,.],.]],.]],.]]
 => [1,5,6,4,7,3,8,2] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 3 - 1
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,3,4,6,7,5,8,2] => [1,8,3,4,7,6,5,2] => ? => ? = 3 - 1
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
 => [1,3,5,4,7,6,8,2] => [1,8,3,5,4,7,6,2] => ? => ? = 3 - 1
[[.,.],[[[[[[.,.],.],.],.],.],.]]
 => [1,3,4,5,6,7,8,2] => [1,8,3,4,5,6,7,2] => ? => ? = 3 - 1
[[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
 => [2,1,5,7,8,6,4,3] => [2,1,8,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 5 - 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: 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: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 45%
Values
[.,.]
 => [1] => [.,.]
 => ([],1)
 => ? = 1
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ? = 2
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2
[[.,.],[.,.]]
 => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3
[[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[[[.,.],.],[.,[.,[.,.]]]]
 => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[[[[.,.],.],.],[.,[.,.]]]
 => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[[[[[.,.],.],.],.],[.,.]]
 => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[[[[.,.],.],.],[.,[.,[.,.]]]]
 => [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[[[[[[[.,.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 45%
Values
[.,.]
 => [1] => [.,.]
 => ([],1)
 => ? = 1 - 1
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ? = 2 - 1
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2 - 1
[[.,.],[.,.]]
 => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2 - 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 - 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 - 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 - 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 - 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 - 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 - 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 - 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 - 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 - 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 - 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 - 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 - 1
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 - 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 - 1
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 - 1
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 - 1
[[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 - 1
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 - 1
[[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 - 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 - 1
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[[[.,.],.],[.,[.,[.,.]]]]
 => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[[[[.,.],.],.],[.,[.,.]]]
 => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[[[[[.,.],.],.],.],[.,.]]
 => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[[[[.,.],.],.],[.,[.,[.,.]]]]
 => [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[[[[[[[.,.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 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.