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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000054
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [1,2] => 1
[[.,.],.]
=> [1,2] => [2,1] => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 3
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => 3
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 4
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => 3
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => 4
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => 4
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => 4
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Values
[.,.]
=> [1,0]
=> [1] => 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ? => ? = 1
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ?
=> ? => ? = 1
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ? => ? = 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11] => ? = 11
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ?
=> ? => ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
=> ?
=> ? => ? = 6
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
=> ?
=> ? => ? = 7
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]
=> ?
=> ? => ? = 8
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? => ? = 9
[.,[[[[[[.,.],.],.],.],.],[[.,.],.]]]
=> ?
=> ? => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[.,.],.]
=> [1,2] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,7,6,5,8,4,3,2] => [2,3,4,8,5,6,7,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,4,7,8,6,5,3,2] => [2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,6,5,4,7,8,3,2] => [2,3,8,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,3,4,5,7,8,6,2] => [2,6,8,7,5,4,3,1] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[[[.,.],.],[.,[[.,[.,[.,.]]],.]]]
=> [1,2,7,6,5,8,4,3] => [3,4,8,5,6,7,2,1] => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,2,4,6,7,8,5,3] => [3,5,8,7,6,4,2,1] => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]
=> [5,4,3,2,1,9,8,7,6] => [6,7,8,9,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> [6,5,4,3,2,1,9,8,7] => [7,8,9,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1,10,9,8,7,6] => [6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
=> [6,5,4,3,2,1,10,9,8,7] => [7,8,9,10,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]
=> [7,6,5,4,3,2,1,10,9,8] => [8,9,10,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 8 + 1
[.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,7,8,10,9,1] => [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 1 + 1
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> [8,7,6,5,4,3,2,10,9,1] => [1,9,10,2,3,4,5,6,7,8] => ?
=> ? = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000645
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [8,7,5,4,2,3,6,1] => [8,5,6,4,3,7,2,1] => ?
=> ? = 1 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [7,8,4,5,2,3,6,1] => [8,5,6,3,4,7,1,2] => ?
=> ? = 1 - 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [7,5,2,3,4,6,8,1] => [8,3,4,5,2,6,1,7] => ?
=> ? = 1 - 1
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [8,6,7,3,2,4,1,5] => [7,5,4,6,8,2,3,1] => ?
=> ? = 5 - 1
[[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [8,7,4,3,1,2,5,6] => [5,6,4,3,7,8,2,1] => ?
=> ? = 6 - 1
[[.,[[.,[[.,.],[.,.]]],.]],[.,.]]
=> [8,5,3,4,2,6,1,7] => [7,5,3,4,2,6,8,1] => ?
=> ? = 7 - 1
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [8,6,5,3,2,1,4,7] => [6,5,4,7,3,2,8,1] => ?
=> ? = 7 - 1
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [8,6,4,2,1,3,5,7] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 7 - 1
[[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [3,4,5,6,2,7,8,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 7 - 1
[[[.,[[.,.],.]],[[[.,.],.],.]],.]
=> [5,6,7,2,3,1,4,8] => [6,4,5,7,1,2,3,8] => ?
=> ? = 8 - 1
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [7,6,4,3,1,2,5,8] => [5,6,4,3,7,2,1,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [6,7,3,4,1,2,5,8] => [5,6,3,4,7,1,2,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [7,6,3,2,1,4,5,8] => [5,4,3,6,7,2,1,8] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [7,5,3,2,1,4,6,8] => [5,4,3,6,2,7,1,8] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 8 - 1
[[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [7,5,1,2,3,4,6,8] => [3,4,5,6,2,7,1,8] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 8 - 1
[[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [7,4,1,2,3,5,6,8] => [3,4,5,2,6,7,1,8] => ?
=> ? = 8 - 1
[[[.,[[[[.,.],.],[.,.]],.]],.],.]
=> [5,2,3,4,6,1,7,8] => [6,2,3,4,1,5,7,8] => ?
=> ? = 8 - 1
[[[[.,.],[[.,[[.,.],.]],.]],.],.]
=> [4,5,3,6,1,2,7,8] => [5,6,3,1,2,4,7,8] => ?
=> ? = 8 - 1
[[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [6,4,2,1,3,5,7,8] => [4,3,5,2,6,1,7,8] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [6,3,1,2,4,5,7,8] => [3,4,2,5,6,1,7,8] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [5,3,1,2,4,6,7,8] => [3,4,2,5,1,6,7,8] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [10,8,6,4,2,1,3,5,7,9] => [6,5,7,4,8,3,9,2,10,1] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 9 - 1
[[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]],.]
=> [8,7,5,4,3,2,1,6,9] => [7,6,5,4,3,8,2,1,9] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> ? = 9 - 1
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => => ? = 1
[.,[.,.]]
=> [2,1] => [1,2] => 1 => 1
[[.,.],.]
=> [1,2] => [2,1] => 0 => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 11 => 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 10 => 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 01 => 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 00 => 3
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 00 => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 111 => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 110 => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 101 => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 100 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 100 => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 011 => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 010 => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 001 => 3
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 001 => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 000 => 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 000 => 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 000 => 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 000 => 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 000 => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 1110 => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => 1101 => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 1100 => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 1100 => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 1011 => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 1010 => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 1001 => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 1001 => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => 1000 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => 1000 => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => 1000 => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => 1000 => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 0111 => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 0110 => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 0101 => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => 0100 => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => 0100 => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => 0011 => 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => 0010 => 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 0011 => 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 0010 => 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => 0001 => 4
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => 0001 => 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => 0001 => 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => 0001 => 4
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 0001 => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => 0000 => 5
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
=> [6,5,3,4,7,8,2,1] => [1,2,8,7,4,3,5,6] => ? => ? = 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [7,8,4,5,2,3,6,1] => [1,6,3,2,5,4,8,7] => ? => ? = 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [5,6,7,2,3,4,8,1] => [1,8,4,3,2,7,6,5] => ? => ? = 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [7,5,2,3,4,6,8,1] => [1,8,6,4,3,2,5,7] => ? => ? = 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [6,7,8,4,5,1,2,3] => [3,2,1,5,4,8,7,6] => ? => ? = 3
[[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [5,6,7,4,8,1,2,3] => [3,2,1,8,4,7,6,5] => ? => ? = 3
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [5,4,6,7,8,1,2,3] => [3,2,1,8,7,6,4,5] => ? => ? = 3
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [8,6,7,3,2,4,1,5] => [5,1,4,2,3,7,6,8] => ? => ? = 5
[[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => [6,3,2,1,4,5,7,8] => ? => ? = 6
[[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [8,7,4,3,1,2,5,6] => [6,5,2,1,3,4,7,8] => ? => ? = 6
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
=> [7,8,2,3,1,4,5,6] => [6,5,4,1,3,2,8,7] => ? => ? = 6
[[.,[[.,[[.,.],[.,.]]],.]],[.,.]]
=> [8,5,3,4,2,6,1,7] => [7,1,6,2,4,3,5,8] => ? => ? = 7
[[.,[[[.,[.,[.,.]]],.],.]],[.,.]]
=> [8,4,3,2,5,6,1,7] => [7,1,6,5,2,3,4,8] => ? => ? = 7
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [6,2,3,4,5,7,1,8] => [8,1,7,5,4,3,2,6] => ? => ? = 8
[[[.,.],[[[.,.],[[.,.],.]],.]],.]
=> [5,6,3,4,7,1,2,8] => [8,2,1,7,4,3,6,5] => ? => ? = 8
[[[.,[[.,.],.]],[[[.,.],.],.]],.]
=> [5,6,7,2,3,1,4,8] => [8,4,1,3,2,7,6,5] => ? => ? = 8
[[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [7,6,3,2,1,4,5,8] => [8,5,4,1,2,3,6,7] => ? => ? = 8
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [7,5,3,2,1,4,6,8] => [8,6,4,1,2,3,5,7] => ? => ? = 8
[[[.,[[[[.,.],.],.],[.,.]]],.],.]
=> [6,2,3,4,5,1,7,8] => [8,7,1,5,4,3,2,6] => ? => ? = 8
[[[.,[[[[.,.],.],[.,.]],.]],.],.]
=> [5,2,3,4,6,1,7,8] => [8,7,1,6,4,3,2,5] => ? => ? = 8
[[[[.,.],[[.,[[.,.],.]],.]],.],.]
=> [4,5,3,6,1,2,7,8] => [8,7,2,1,6,3,5,4] => ? => ? = 8
[[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [6,4,2,1,3,5,7,8] => [8,7,5,3,1,2,4,6] => ? => ? = 8
[[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [6,3,1,2,4,5,7,8] => [8,7,5,4,2,1,3,6] => ? => ? = 8
[[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]],.]
=> [8,7,5,4,3,2,1,6,9] => [9,6,1,2,3,4,5,7,8] => ? => ? = 9
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 82%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 82%
Values
[.,.]
=> [1,0]
=> [1] => [1] => 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [2] => [2] => 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => [3] => 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => [3] => 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,2,1] => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1] => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [1,3] => 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4] => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4] => 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4] => 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4] => 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,3,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,3,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [2,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,4] => 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,4] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,4] => 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,4] => 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => 4
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,1,4] => [1,1,4,2] => ? = 2
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,2] => [4,2,2] => ? = 2
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,6] => [6,2] => ? = 2
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6] => [6,2] => ? = 2
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
[[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,5] => [5,3] => ? = 3
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,5] => [5,3] => ? = 3
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [5,1,1,1] => [1,1,1,5] => ? = 5
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [5,1,2] => [1,2,5] => ? = 5
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [5,3] => [3,5] => ? = 5
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [5,2,1] => [2,1,5] => ? = 5
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => [3,5] => ? = 5
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [5,1,1,1] => [1,1,1,5] => ? = 5
[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [5,2,1] => [2,1,5] => ? = 5
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [5,3] => [3,5] => ? = 5
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,1,1] => [1,1,6] => ? = 6
[[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [6,1,1] => [1,1,6] => ? = 6
[[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [6,1,1] => [1,1,6] => ? = 6
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [6,1,1] => [1,1,6] => ? = 6
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [6,2] => [2,6] => ? = 6
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8] => [8] => ? = 8
[[.,[.,[[[.,.],.],[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [8] => [8] => ? = 8
[[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [8] => [8] => ? = 8
[[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [8] => [8] => ? = 8
[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[.,[[[[[.,.],.],.],.],[.,.]]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [8] => [8] => ? = 8
[[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[.,.],[[[.,.],.],[[.,.],.]]],.]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [8] => [8] => ? = 8
[[[.,.],[[[.,.],[[.,.],.]],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [8] => [8] => ? = 8
[[[.,.],[[[.,[[.,.],.]],.],.]],.]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[.,.],.],[[.,[[.,.],.]],.]],.]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [8] => [8] => ? = 8
[[[.,[[.,.],.]],[[[.,.],.],.]],.]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [8] => [8] => ? = 8
[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [8] => [8] => ? = 8
[[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [8] => [8] => ? = 8
[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [8] => [8] => ? = 8
[[[.,[[[[.,.],.],.],[.,.]]],.],.]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [8] => [8] => ? = 8
[[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[.,[[[[.,.],.],[.,.]],.]],.],.]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [8] => [8] => ? = 8
Description
The last part of an integer composition.
Matching statistic: St000505
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 91%
Mp00064: Permutations —reverse⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 91%
Values
[.,.]
=> [1] => [1] => {{1}}
=> 1
[.,[.,.]]
=> [2,1] => [1,2] => {{1},{2}}
=> 1
[[.,.],.]
=> [1,2] => [2,1] => {{1,2}}
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => {{1},{2},{3}}
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => {{1},{2,3}}
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => {{1,2},{3}}
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => {{1,2,3}}
=> 3
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => {{1,3},{2}}
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => {{1,2,3},{4}}
=> 3
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => {{1,2,3,4}}
=> 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => {{1,2,4},{3}}
=> 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => {{1,3,4},{2}}
=> 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => {{1,2,3,4}}
=> 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => {{1,4},{2,3}}
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => {{1},{2,3,4,5}}
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 4
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => {{1,2,4},{3},{5}}
=> 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => {{1,3,4},{2},{5}}
=> 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => {{1,2,3,4},{5}}
=> 4
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [8,7,6,4,5,3,2,1] => [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [8,7,3,4,5,6,2,1] => [1,2,6,5,4,3,7,8] => {{1},{2},{3,6},{4,5},{7},{8}}
=> ? = 1
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
=> [6,5,3,4,7,8,2,1] => [1,2,8,7,4,3,5,6] => ?
=> ? = 1
[.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
=> [5,4,3,6,7,8,2,1] => [1,2,8,7,6,3,4,5] => {{1},{2},{3,5,6,8},{4,7}}
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [8,6,7,4,5,2,3,1] => [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[.,[[.,[.,.]],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,2,4,1] => [1,4,2,3,8,6,5,7] => {{1},{2,3,4},{5,7,8},{6}}
=> ? = 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => {{1},{2,5},{3,4},{6,8},{7}}
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => [1,6,2,3,4,5,7,8] => {{1},{2,3,4,5,6},{7},{8}}
=> ? = 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [8,7,5,4,2,3,6,1] => [1,6,3,2,4,5,7,8] => {{1},{2,4,5,6},{3},{7},{8}}
=> ? = 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [7,8,4,5,2,3,6,1] => [1,6,3,2,5,4,8,7] => ?
=> ? = 1
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [8,7,2,3,4,5,6,1] => [1,6,5,4,3,2,7,8] => {{1},{2,6},{3,5},{4},{7},{8}}
=> ? = 1
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => [1,6,5,4,3,2,8,7] => {{1},{2,6},{3,5},{4},{7,8}}
=> ? = 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => [1,7,6,5,4,3,2,8] => {{1},{2,7},{3,6},{4,5},{8}}
=> ? = 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => {{1},{2,3,5,8},{4,7},{6}}
=> ? = 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [5,6,7,2,3,4,8,1] => [1,8,4,3,2,7,6,5] => {{1},{2,5,8},{3,4},{6,7}}
=> ? = 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [7,5,2,3,4,6,8,1] => [1,8,6,4,3,2,5,7] => ?
=> ? = 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [7,2,3,4,5,6,8,1] => [1,8,6,5,4,3,2,7] => {{1},{2,7,8},{3,6},{4,5}}
=> ? = 1
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [6,5,2,3,4,7,8,1] => [1,8,7,4,3,2,5,6] => {{1},{2,6,8},{3,5,7},{4}}
=> ? = 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [6,2,3,4,5,7,8,1] => [1,8,7,5,4,3,2,6] => {{1},{2,3,6,7,8},{4,5}}
=> ? = 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => {{1},{2,3,7,8},{4,6},{5}}
=> ? = 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
=> ? = 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,3,1,2] => [2,1,3,4,8,5,6,7] => {{1,2},{3},{4},{5,6,7,8}}
=> ? = 2
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [7,8,6,4,5,3,1,2] => [2,1,3,5,4,6,8,7] => {{1,2},{3},{4,5},{6},{7,8}}
=> ? = 2
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> [6,5,4,7,8,3,1,2] => [2,1,3,8,7,4,5,6] => {{1,2},{3},{4,6,8},{5,7}}
=> ? = 2
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [6,7,3,4,5,8,1,2] => [2,1,8,5,4,3,7,6] => {{1,2},{3,6,8},{4,5},{7}}
=> ? = 2
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [4,3,5,6,7,8,1,2] => [2,1,8,7,6,5,3,4] => {{1,2},{3,4,7,8},{5,6}}
=> ? = 2
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 3
[[[.,.],.],[.,[[.,[.,[.,.]]],.]]]
=> [7,6,5,8,4,1,2,3] => [3,2,1,4,8,5,6,7] => {{1,3},{2},{4},{5,6,7,8}}
=> ? = 3
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [6,7,8,4,5,1,2,3] => [3,2,1,5,4,8,7,6] => ?
=> ? = 3
[[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [5,6,7,4,8,1,2,3] => [3,2,1,8,4,7,6,5] => {{1,3},{2},{4,5,8},{6,7}}
=> ? = 3
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [5,4,6,7,8,1,2,3] => [3,2,1,8,7,6,4,5] => {{1,3},{2},{4,5,7,8},{6}}
=> ? = 3
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,1,2,3,4] => [4,3,2,1,8,7,5,6] => {{1,4},{2,3},{5,6,7,8}}
=> ? = 4
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [7,8,6,3,4,2,1,5] => [5,1,2,4,3,6,8,7] => {{1,2,3,5},{4},{6},{7,8}}
=> ? = 5
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [7,6,8,4,2,3,1,5] => [5,1,3,2,4,8,6,7] => {{1,2,4,5},{3},{6,7,8}}
=> ? = 5
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [8,6,7,3,2,4,1,5] => [5,1,4,2,3,7,6,8] => ?
=> ? = 5
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [6,7,8,2,3,4,1,5] => [5,1,4,3,2,8,7,6] => {{1,2,5},{3,4},{6,8},{7}}
=> ? = 5
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [8,6,7,4,2,1,3,5] => [5,3,1,2,4,7,6,8] => {{1,2,3,4,5},{6,7},{8}}
=> ? = 5
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [7,6,8,1,2,3,4,5] => [5,4,3,2,1,8,6,7] => {{1,5},{2,4},{3},{6,7,8}}
=> ? = 5
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [8,7,5,4,3,2,1,6] => [6,1,2,3,4,5,7,8] => {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => [6,3,2,1,4,5,7,8] => ?
=> ? = 6
[[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [8,7,4,3,1,2,5,6] => [6,5,2,1,3,4,7,8] => ?
=> ? = 6
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [8,7,3,2,1,4,5,6] => [6,5,4,1,2,3,7,8] => {{1,3,4,6},{2,5},{7},{8}}
=> ? = 6
Description
The biggest entry in the block containing the 1.
Matching statistic: St000476
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 73%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 73%
Values
[.,.]
=> [1] => [1] => [1,0]
=> ? = 1 - 1
[.,[.,.]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [8,7,6,1,5,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [8,7,6,4,2,1,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [3,4,5,8,7,6,2,1] => [8,7,1,2,3,6,5,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
=> [3,6,5,4,7,8,2,1] => [8,7,1,4,3,2,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
=> [5,4,3,6,7,8,2,1] => [8,7,3,2,1,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [2,4,6,8,7,5,3,1] => [8,1,7,2,6,3,5,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[.,[.,.]],[[[.,.],[.,.]],.]]]
=> [3,2,5,7,6,8,4,1] => [8,2,1,7,3,5,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [2,3,4,6,7,8,5,1] => [8,1,2,3,7,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [5,4,3,2,8,7,6,1] => [8,4,3,2,1,7,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [2,5,4,3,8,7,6,1] => [8,1,4,3,2,7,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [2,4,5,3,7,8,6,1] => [8,1,4,2,3,7,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [2,3,4,5,8,7,6,1] => [8,1,2,3,4,7,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,7,8,6,1] => [8,1,2,3,4,7,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [6,5,4,3,2,8,7,1] => [8,5,4,3,2,1,7,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [2,4,3,5,6,8,7,1] => [8,1,3,2,4,5,7,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,8,7,1] => [8,1,2,3,4,5,7,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [8,6,5,4,2,1,3,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [8,6,4,3,2,1,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [8,6,4,2,1,3,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [2,3,5,6,7,4,8,1] => [8,1,2,6,3,4,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [2,3,5,4,7,6,8,1] => [8,1,2,4,3,6,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [2,3,4,5,7,6,8,1] => [8,1,2,3,4,6,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [2,3,6,5,4,7,8,1] => [8,1,2,5,4,3,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [2,3,5,6,4,7,8,1] => [8,1,2,5,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [2,3,4,6,5,7,8,1] => [8,1,2,3,5,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [3,4,2,1,7,8,6,5] => [4,3,1,2,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [2,4,3,1,7,6,8,5] => [4,1,3,2,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [3,2,4,1,6,8,7,5] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [2,3,4,1,6,7,8,5] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
=> [2,3,1,4,5,7,8,6] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[.,[[.,[[.,.],[.,.]]],.]],[.,.]]
=> [3,5,4,2,6,1,8,7] => [6,4,1,3,2,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[.,[[[.,[.,[.,.]]],.],.]],[.,.]]
=> [4,3,2,5,6,1,8,7] => [6,3,2,1,4,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [3,2,1,6,5,4,8,7] => [3,2,1,6,5,4,8,7] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[[.,[.,[[[.,.],.],[.,[.,.]]]]],.]
=> [3,4,7,6,5,2,1,8] => [7,6,1,2,5,4,3,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> [5,6,4,3,7,2,1,8] => [7,6,4,3,1,2,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> [5,4,3,6,7,2,1,8] => [7,6,3,2,1,4,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000026
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 64%
St000026: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1,0]
=> 1
[.,[.,.]]
=> [1,0,1,0]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 1
[.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[.,[[.,[.,.]],[[[.,.],[.,.]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 1
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[[[.,.],.],[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 3
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 5
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 5
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
Description
The position of the first return of a Dyck path.
Matching statistic: St000025
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 64%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[.,.],.]
=> [1,2] => [2,1] => [1,1,0,0]
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [3,4,5,8,7,6,2,1] => [1,2,6,7,8,5,4,3] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
=> [3,6,5,4,7,8,2,1] => [1,2,8,7,4,5,6,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
=> [5,4,3,6,7,8,2,1] => [1,2,8,7,6,3,4,5] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [2,4,6,8,7,5,3,1] => [1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[.,[[.,[.,.]],[[[.,.],[.,.]],.]]]
=> [3,2,5,7,6,8,4,1] => [1,4,8,6,7,5,2,3] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [2,3,4,6,7,8,5,1] => [1,5,8,7,6,4,3,2] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [5,4,3,2,8,7,6,1] => [1,6,7,8,2,3,4,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [2,5,4,3,8,7,6,1] => [1,6,7,8,3,4,5,2] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [2,4,5,3,7,8,6,1] => [1,6,8,7,3,5,4,2] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [2,3,4,5,8,7,6,1] => [1,6,7,8,5,4,3,2] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,7,8,6,1] => [1,6,8,7,5,4,3,2] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [6,5,4,3,2,8,7,1] => [1,7,8,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [2,4,3,5,6,8,7,1] => [1,7,8,6,5,3,4,2] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,8,7,1] => [1,7,8,6,5,4,3,2] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [1,8,2,7,3,4,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [2,3,5,6,7,4,8,1] => [1,8,4,7,6,5,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [2,3,5,4,7,6,8,1] => [1,8,6,7,4,5,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [2,3,4,5,7,6,8,1] => [1,8,6,7,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [2,3,6,5,4,7,8,1] => [1,8,7,4,5,6,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [2,3,5,6,4,7,8,1] => [1,8,7,4,6,5,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [2,3,4,6,5,7,8,1] => [1,8,7,5,6,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,7,6,5,8,4,3,2] => [2,3,4,8,5,6,7,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,4,7,8,6,5,3,2] => [2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,6,5,4,7,8,3,2] => [2,3,8,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,3,4,5,7,8,6,2] => [2,6,8,7,5,4,3,1] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [1,3,4,6,7,5,8,2] => [2,8,5,7,6,4,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [1,4,3,5,6,7,8,2] => [2,8,7,6,5,3,4,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [2,1,8,7,6,5,4,3] => [3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3
[[[.,.],.],[.,[[.,[.,[.,.]]],.]]]
=> [1,2,7,6,5,8,4,3] => [3,4,8,5,6,7,2,1] => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,2,4,6,7,8,5,3] => [3,5,8,7,6,4,2,1] => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3
[[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [1,2,5,6,7,4,8,3] => [3,8,4,7,6,5,2,1] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [1,2,5,4,6,7,8,3] => [3,8,7,6,4,5,2,1] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,2,3,6,5,7,8,4] => [4,8,7,5,6,3,2,1] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [4,3,2,1,8,7,6,5] => [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 5
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [3,4,2,1,7,8,6,5] => [5,6,8,7,1,2,4,3] => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 5
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [2,4,3,1,7,6,8,5] => [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [3,2,4,1,6,8,7,5] => [5,7,8,6,1,4,2,3] => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 5
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [2,3,4,1,6,7,8,5] => [5,8,7,6,1,4,3,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000171The degree of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001725The harmonious chromatic number of a graph. St000501The size of the first part in the decomposition of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000740The last entry of a permutation. St000051The size of the left subtree of a binary tree. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. St000133The "bounce" of a permutation. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001882The number of occurrences of a type-B 231 pattern in a signed permutation.
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