Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000026
(load all 16 compositions to match this statistic)
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The position of the first return of a Dyck path.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
Mapping
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The first part of an integer composition.
Matching statistic: St000501
(load all 53 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000501: 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] => [1,3,2] => 1
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => 1
[[.,.],[.,.]]
 => [1,3,2] => [2,1,3] => 2
[[.,[.,.]],.]
 => [2,1,3] => [3,2,1] => 3
[[[.,.],.],.]
 => [1,2,3] => [2,3,1] => 3
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,4,3,2] => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,4,2,3] => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,2,4,3] => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,3,2,4] => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,4,3] => 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [2,1,3,4] => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [3,2,1,4] => 3
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [2,3,1,4] => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,3,2,1] => 4
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,2,3,1] => 4
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [2,4,3,1] => 4
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [3,2,4,1] => 4
[[[[.,.],.],.],.]
 => [1,2,3,4] => [2,3,4,1] => 4
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,5,4,3,2] => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,5,4,2,3] => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [1,5,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,5,3,2,4] => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,5,2,3,4] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [1,2,5,4,3] => 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [1,2,5,3,4] => 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [1,3,2,5,4] => 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [1,2,3,5,4] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,4,3,2,5] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,4,2,3,5] => 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [1,2,4,3,5] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,3,2,4,5] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,1,5,4,3] => 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [2,1,5,3,4] => 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [2,1,3,5,4] => 2
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [2,1,4,3,5] => 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [2,1,3,4,5] => 2
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [3,2,1,5,4] => 3
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,2,1,4,5] => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [2,3,1,5,4] => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [2,3,1,4,5] => 3
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [4,3,2,1,5] => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [4,2,3,1,5] => 4
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [2,4,3,1,5] => 4
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [3,2,4,1,5] => 4
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [2,3,4,1,5] => 4
Description
The size of the first part in the decomposition of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is defined to be the smallest $k > 0$ such that $\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$. This statistic is undefined for the empty permutation. For the number of parts in the decomposition see [[St000056]].
Matching statistic: St000505
(load all 8 compositions to match this statistic)
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => {{1}}
 => 1
[.,[.,.]]
 => [1,0,1,0]
 => {{1},{2}}
 => 1
[[.,.],.]
 => [1,1,0,0]
 => {{1,2}}
 => 2
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 3
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 3
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 3
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 3
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 4
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 4
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 4
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 4
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 4
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 3
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 4
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 4
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 4
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 4
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 4
Description
The biggest entry in the block containing the 1.
Matching statistic: St000025
(load all 2 compositions to match this statistic)
Mapping
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000054
(load all 29 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 1
[.,[.,.]]
 => [2,1] => [1,2] => [1,2] => 1
[[.,.],.]
 => [1,2] => [2,1] => [2,1] => 2
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,3,2] => 1
[.,[[.,.],.]]
 => [2,3,1] => [1,3,2] => [1,3,2] => 1
[[.,.],[.,.]]
 => [3,1,2] => [2,1,3] => [2,1,3] => 2
[[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => [3,1,2] => 3
[[[.,.],.],.]
 => [1,2,3] => [3,2,1] => [3,2,1] => 3
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,4,3] => [1,4,3,2] => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [2,1,3,4] => [2,1,4,3] => 2
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,1,2,4] => [3,1,4,2] => 3
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 4
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 4
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 4
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 4
[[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,5,4] => [1,5,4,3,2] => 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,4,3,5] => [1,5,4,3,2] => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,5,3,4] => [1,5,4,3,2] => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,5,4,3] => [1,5,4,3,2] => 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,3,2,4,5] => [1,5,4,3,2] => 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,3,2,5,4] => [1,5,4,3,2] => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,4,2,3,5] => [1,5,4,3,2] => 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,5,2,3,4] => [1,5,4,3,2] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,5,2,4,3] => [1,5,4,3,2] => 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,5,3,2,4] => [1,5,4,3,2] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [2,1,3,4,5] => [2,1,5,4,3] => 2
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [2,1,3,5,4] => [2,1,5,4,3] => 2
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [3,1,2,4,5] => [3,1,5,4,2] => 3
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,1,2,5,4] => [3,1,5,4,2] => 3
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,5,4] => 3
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 3
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,1,2,3,5] => [4,1,5,3,2] => 4
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [4,1,3,2,5] => [4,1,5,3,2] => 4
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,5,3] => 4
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,5,2] => 4
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 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: St000383
(load all 2 compositions to match this statistic)
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1,0]
 => [1] => 1
[.,[.,.]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1] => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,1,0,0]
 => [2] => 2
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1] => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [3] => 3
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [3] => 3
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 3
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => 4
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [4] => 4
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => 4
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => 4
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 4
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [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]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 2
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 2
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 3
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 3
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 3
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 4
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 4
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 4
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 4
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 4
Description
The last part of an integer composition.
Matching statistic: St001725
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1] => ([],1)
 => 1
[.,[.,.]]
 => [1,1,0,0]
 => [2] => ([],2)
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 2
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 2
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 4
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 4
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => ([],5)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => ([],5)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => ([],5)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => ([],5)
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000171
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1] => ([],1)
 => 0 = 1 - 1
[.,[.,.]]
 => [1,1,0,0]
 => [2] => ([],2)
 => 0 = 1 - 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => 0 = 1 - 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
Mapping
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: 100% ●values known / values provided: 100%●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
Description
The position of the first down step of a Dyck path.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000051The size of the left subtree of a binary tree. St000740The last entry of a permutation. 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.