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Your data matches 31 different statistics following compositions of up to 3 maps.
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Matching statistic: St000366
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000371
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000931
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[.,.]
=> [.,.]
=> [1,0]
=> ? = 0
[.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> 0
[[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 2
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 0
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],.]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [[.,[[.,.],[[.,[[.,.],.]],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [[.,[[.,.],[[.,.],[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [[.,[[.,[[.,.],.]],.]],[[.,.],.]]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [[.,[[.,.],[[.,.],.]]],[[.,.],.]]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 0
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000118
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St000118: Binary trees ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Values
[.,.]
=> 0
[.,[.,.]]
=> 0
[[.,.],.]
=> 0
[.,[.,[.,.]]]
=> 1
[.,[[.,.],.]]
=> 0
[[.,.],[.,.]]
=> 0
[[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> 2
[.,[.,[[.,.],.]]]
=> 1
[.,[[.,.],[.,.]]]
=> 1
[.,[[.,[.,.]],.]]
=> 0
[.,[[[.,.],.],.]]
=> 0
[[.,.],[.,[.,.]]]
=> 1
[[.,.],[[.,.],.]]
=> 0
[[.,[.,.]],[.,.]]
=> 0
[[[.,.],.],[.,.]]
=> 0
[[.,[.,[.,.]]],.]
=> 1
[[.,[[.,.],.]],.]
=> 0
[[[.,.],[.,.]],.]
=> 0
[[[.,[.,.]],.],.]
=> 0
[[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> 3
[.,[.,[.,[[.,.],.]]]]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> 2
[.,[[.,.],[[.,.],.]]]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> 0
[.,[[[[.,.],.],.],.]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> 2
[[.,.],[.,[[.,.],.]]]
=> 1
[[.,.],[[.,.],[.,.]]]
=> 1
[[.,.],[[.,[.,.]],.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> 1
[[[.,.],.],[[.,.],.]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> 0
[[[[.,.],.],.],[.,.]]
=> 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> ? = 0
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> ? = 0
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> ? = 0
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ? = 0
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> ? = 0
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 5
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 4
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> ? = 0
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> ? = 0
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> ? = 0
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 4
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 0
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> ? = 2
[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> ? = 2
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> ? = 0
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> ? = 0
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 4
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 4
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> ? = 0
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> ? = 0
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> ? = 2
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> ? = 0
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> ? = 2
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> ? = 0
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> ? = 0
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> ? = 0
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 5
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> ? = 2
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> ? = 0
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> ? = 0
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ? = 2
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St001066
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Values
[.,.]
=> [.,.]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[[.,.],.]
=> [[.,.],.]
=> [1,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [.,[[.,.],[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [.,[[[.,.],[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [[.,[[.,.],[[.,.],[.,.]]]],[.,.]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [[.,[[[.,.],[.,.]],[.,.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [[[.,[[.,.],[.,.]]],[.,.]],[.,.]]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [[[.,[.,.]],[[.,.],[.,.]]],[.,.]]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001167
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 0
[.,[.,.]]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0
[[.,.],.]
=> [1,2] => [2] => [1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[[.,.],[.,.]]
=> [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[[[.,.],.],.]
=> [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [2,1,8,7,6,5,4,3] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,6,5,7,4,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [2,1,4,5,6,7,8,3] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [3,2,1,8,7,6,5,4] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,5,6,7,8,4] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [4,3,2,1,8,7,6,5] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [4,3,2,1,7,6,8,5] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [3,2,4,1,8,7,6,5] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,4,1,7,6,8,5] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,7,6,8,5] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [4,3,5,2,6,1,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,6,1,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [2,1,6,5,4,3,8,7] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [2,1,5,4,6,3,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [4,3,2,1,6,5,8,7] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,8,7] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [4,3,2,1,5,6,7,8] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 0
[.,[.,.]]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0
[[.,.],.]
=> [1,2] => [2] => [1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[[.,.],[.,.]]
=> [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[[[.,.],.],.]
=> [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [2,1,8,7,6,5,4,3] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,6,5,7,4,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [2,1,4,5,6,7,8,3] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [3,2,1,8,7,6,5,4] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,5,6,7,8,4] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [4,3,2,1,8,7,6,5] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [4,3,2,1,7,6,8,5] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [3,2,4,1,8,7,6,5] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,4,1,7,6,8,5] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,7,6,8,5] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [4,3,5,2,6,1,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,6,1,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [2,1,6,5,4,3,8,7] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [2,1,5,4,6,3,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [4,3,2,1,6,5,8,7] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,8,7] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [4,3,2,1,5,6,7,8] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St001483
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Values
[.,.]
=> [.,.]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],.]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 0 + 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [[.,[[.,.],[[.,[[.,.],.]],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 0 + 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [[.,[[.,.],[[.,.],[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 0 + 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 + 1
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [[.,[[.,[[.,.],.]],.]],[[.,.],.]]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 0 + 1
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [[.,[[.,.],[[.,.],.]]],[[.,.],.]]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 4 + 1
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [[.,.],[[.,[[.,[[.,.],.]],.]],.]]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [[.,.],[[.,[[.,.],[[.,.],.]]],.]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5 + 1
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000358
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 89%●distinct values known / distinct values provided: 71%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 89%●distinct values known / distinct values provided: 71%
Values
[.,.]
=> [1,0]
=> [[1]]
=> [1] => 0
[.,[.,.]]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 0
[[.,.],.]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 2
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => 3
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [7,1,2,3,4,5,6] => ? = 5
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[1,0,0,0,0,-1,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,-1,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,-1,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,-1,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [5,1,2,3,4,7,6] => ? = 3
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [5,1,2,3,4,7,6] => ? = 3
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [6,1,2,3,4,5,7] => ? = 4
[[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [5,1,2,3,4,6,7] => ? = 3
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [4,1,2,3,5,6,7] => ? = 2
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [3,1,2,4,5,6,7] => ? = 1
[[[[[[.,[.,.]],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,5,6,7] => ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [8,1,2,3,4,5,6,7] => ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,-1,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ? = 0
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,8,2,3,4,5,6,7] => ? = 5
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,8,3,4,5,6,7] => ? = 4
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,8,7] => ? = 0
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [3,1,2,8,4,5,6,7] => ? = 4
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ? = 0
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [4,1,2,3,8,5,6,7] => ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,4,3,8,5,6,7] => ? = 2
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [5,1,2,3,4,8,6,7] => ? = 4
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [6,1,2,3,4,5,8,7] => ? = 4
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,8,7] => ? = 0
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001727
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 89%●distinct values known / distinct values provided: 71%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 89%●distinct values known / distinct values provided: 71%
Values
[.,.]
=> [1,0]
=> [[1]]
=> [1] => 0
[.,[.,.]]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 0
[[.,.],.]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 2
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => 3
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [7,1,2,3,4,5,6] => ? = 5
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[1,0,0,0,0,-1,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,-1,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,-1,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,-1,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [5,1,2,3,4,7,6] => ? = 3
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,7,2,3,4,5,6] => ? = 4
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,7,3,4,5,6] => ? = 3
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [5,1,2,3,4,7,6] => ? = 3
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [6,1,2,3,4,5,7] => ? = 4
[[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [5,1,2,3,4,6,7] => ? = 3
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [4,1,2,3,5,6,7] => ? = 2
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [3,1,2,4,5,6,7] => ? = 1
[[[[[[.,[.,.]],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,5,6,7] => ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [8,1,2,3,4,5,6,7] => ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,-1,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ? = 0
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,8,2,3,4,5,6,7] => ? = 5
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,8,3,4,5,6,7] => ? = 4
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,8,7] => ? = 0
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [3,1,2,8,4,5,6,7] => ? = 4
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ? = 0
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [4,1,2,3,8,5,6,7] => ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,4,3,8,5,6,7] => ? = 2
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [5,1,2,3,4,8,6,7] => ? = 4
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [6,1,2,3,4,5,8,7] => ? = 4
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 2
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? = 0
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,8,7] => ? = 0
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000836The number of descents of distance 2 of a permutation. St000732The number of double deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000365The number of double ascents of a permutation. St000223The number of nestings in the permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St001866The nesting alignments of a signed permutation. St001862The number of crossings of a signed permutation. St000982The length of the longest constant subword.
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