searching the database
Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000505
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
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%
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: 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
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [12] => ? = 12
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [12] => ? = 12
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [12] => ? = 12
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [12] => ? = 12
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [12] => ? = 12
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [12] => ? = 12
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [12] => ? = 12
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [12] => ? = 12
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [10,2] => ? = 10
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [10,2] => ? = 10
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [10,2] => ? = 10
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [10,2] => ? = 10
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [8,4] => ? = 8
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [8,2,2] => ? = 8
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,8,2] => ? = 2
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,2,6,2] => ? = 2
[[.,.],[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,2,2,2,4] => ? = 2
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)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
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
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 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
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 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
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 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
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [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
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 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
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 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
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [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
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[[[[.,.],[[[.,.],.],.]],.],.],.]
=> [3,4,5,1,2,6,7,8] => [8,7,6,2,1,5,4,3] => ?
=> ? = 8 + 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [9,10,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 1
[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
=> [9,2,3,1,4,5,6,7,8] => [8,7,6,5,4,1,3,2,9] => ?
=> ? = 8 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [9,7,8,5,6,3,1,2,4,10] => [10,4,2,1,3,6,5,8,7,9] => ?
=> ? = 10 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [9,7,8,5,3,4,1,2,6,10] => [10,6,2,1,4,3,5,8,7,9] => ?
=> ? = 10 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [9,7,8,5,3,4,6,1,2,10] => [10,2,1,6,4,3,5,8,7,9] => ?
=> ? = 10 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [9,10,7,5,6,3,1,2,4,8] => [8,4,2,1,3,6,5,7,10,9] => ?
=> ? = 8 + 1
[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [9,10,7,5,6,3,4,1,2,8] => [8,2,1,4,3,6,5,7,10,9] => ?
=> ? = 8 + 1
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [9,10,7,5,6,3,4,8,1,2] => [2,1,8,4,3,6,5,7,10,9] => ?
=> ? = 2 + 1
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [9,10,7,5,6,8,3,1,2,4] => [4,2,1,3,8,6,5,7,10,9] => ?
=> ? = 4 + 1
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [9,10,7,5,6,8,3,4,1,2] => [2,1,4,3,8,6,5,7,10,9] => ?
=> ? = 2 + 1
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [11,9,10,7,5,6,3,4,1,2,8,12] => [12,8,2,1,4,3,6,5,7,10,9,11] => ?
=> ? = 12 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [11,9,10,7,8,5,3,4,1,2,6,12] => [12,6,2,1,4,3,5,8,7,10,9,11] => ?
=> ? = 12 + 1
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [11,9,10,7,5,6,8,3,1,2,4,12] => [12,4,2,1,3,8,6,5,7,10,9,11] => ?
=> ? = 12 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,8,5,6,3,1,2,4,12] => [12,4,2,1,3,6,5,8,7,10,9,11] => ?
=> ? = 12 + 1
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [11,9,10,7,5,6,3,4,8,1,2,12] => [12,2,1,8,4,3,6,5,7,10,9,11] => ?
=> ? = 12 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,8,5,3,4,6,1,2,12] => [12,2,1,6,4,3,5,8,7,10,9,11] => ?
=> ? = 12 + 1
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,5,6,8,3,4,1,2,12] => [12,2,1,4,3,8,6,5,7,10,9,11] => ?
=> ? = 12 + 1
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [11,9,10,7,8,5,6,3,4,1,2,12] => [12,2,1,4,3,6,5,8,7,10,9,11] => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [11,12,9,7,8,5,3,4,1,2,6,10] => [10,6,2,1,4,3,5,8,7,9,12,11] => ?
=> ? = 10 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [11,12,9,7,8,5,6,3,1,2,4,10] => [10,4,2,1,3,6,5,8,7,9,12,11] => ?
=> ? = 10 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [11,12,9,7,8,5,3,4,6,1,2,10] => [10,2,1,6,4,3,5,8,7,9,12,11] => ?
=> ? = 10 + 1
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [11,12,9,7,8,5,6,3,4,1,2,10] => [10,2,1,4,3,6,5,8,7,9,12,11] => ?
=> ? = 10 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [11,9,10,12,7,5,6,3,1,2,4,8] => [8,4,2,1,3,6,5,7,12,10,9,11] => ?
=> ? = 8 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [11,12,9,10,7,5,6,3,1,2,4,8] => [8,4,2,1,3,6,5,7,10,9,12,11] => ?
=> ? = 8 + 1
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [11,12,9,7,8,5,6,3,4,10,1,2] => [2,1,10,4,3,6,5,8,7,9,12,11] => ?
=> ? = 2 + 1
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [11,12,9,7,8,10,5,3,4,1,2,6] => [6,2,1,4,3,5,10,8,7,9,12,11] => ?
=> ? = 6 + 1
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [11,12,9,7,8,5,6,10,3,4,1,2] => [2,1,4,3,10,6,5,8,7,9,12,11] => ?
=> ? = 2 + 1
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [11,12,9,7,8,5,6,10,3,1,2,4] => [4,2,1,3,10,6,5,8,7,9,12,11] => ?
=> ? = 4 + 1
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [11,12,9,7,8,10,5,6,3,1,2,4] => [4,2,1,3,6,5,10,8,7,9,12,11] => ?
=> ? = 4 + 1
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [11,12,9,7,8,10,5,3,4,6,1,2] => [2,1,6,4,3,5,10,8,7,9,12,11] => ?
=> ? = 2 + 1
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [11,12,9,7,8,10,5,6,3,4,1,2] => [2,1,4,3,6,5,10,8,7,9,12,11] => ?
=> ? = 2 + 1
[[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]],.]
=> [9,7,8,5,3,1,2,4,6,10] => ? => ?
=> ? = 10 + 1
[[[[.,.],[.,.]],[.,.]],[[.,.],[[.,.],.]]]
=> [9,10,7,8,5,3,1,2,4,6] => ? => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[.,.]],[[[.,.],[.,.]],.]]
=> [9,7,8,10,5,3,1,2,4,6] => ? => ?
=> ? = 6 + 1
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],.]]]]
=> [9,7,8,10,5,6,3,4,1,2] => [2,1,4,3,6,5,10,8,7,9] => ?
=> ? = 2 + 1
[[.,.],[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],.]]]]]
=> [11,9,10,12,7,8,5,6,3,4,1,2] => ? => ?
=> ? = 2 + 1
Description
The position of the first down step of a Dyck path.
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: 82% ●values known / values provided: 95%●distinct values known / distinct values provided: 82%
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 95%●distinct values known / distinct values provided: 82%
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
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [8,2,3,1,4,5,6,7] => [7,6,5,4,1,3,2,8] => ? => ? = 7
[[[[[.,.],[[[.,.],.],.]],.],.],.]
=> [3,4,5,1,2,6,7,8] => [8,7,6,2,1,5,4,3] => ? => ? = 8
[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
=> [9,2,3,1,4,5,6,7,8] => [8,7,6,5,4,1,3,2,9] => ? => ? = 8
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [9,7,8,5,6,3,1,2,4,10] => [10,4,2,1,3,6,5,8,7,9] => ? => ? = 10
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [9,7,8,5,3,4,1,2,6,10] => [10,6,2,1,4,3,5,8,7,9] => ? => ? = 10
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [9,7,8,5,3,4,6,1,2,10] => [10,2,1,6,4,3,5,8,7,9] => ? => ? = 10
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [9,7,8,5,6,3,4,1,2,10] => [10,2,1,4,3,6,5,8,7,9] => ? => ? = 10
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [9,10,7,5,6,3,1,2,4,8] => [8,4,2,1,3,6,5,7,10,9] => ? => ? = 8
[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [9,10,7,5,6,3,4,1,2,8] => [8,2,1,4,3,6,5,7,10,9] => ? => ? = 8
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [9,10,7,5,6,3,4,8,1,2] => [2,1,8,4,3,6,5,7,10,9] => ? => ? = 2
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [9,10,7,5,6,8,3,1,2,4] => [4,2,1,3,8,6,5,7,10,9] => ? => ? = 4
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [9,10,7,5,6,8,3,4,1,2] => [2,1,4,3,8,6,5,7,10,9] => ? => ? = 2
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [11,9,10,7,5,6,3,4,1,2,8,12] => [12,8,2,1,4,3,6,5,7,10,9,11] => ? => ? = 12
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [11,9,10,7,8,5,3,4,1,2,6,12] => [12,6,2,1,4,3,5,8,7,10,9,11] => ? => ? = 12
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [11,9,10,7,5,6,8,3,1,2,4,12] => [12,4,2,1,3,8,6,5,7,10,9,11] => ? => ? = 12
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,8,5,6,3,1,2,4,12] => [12,4,2,1,3,6,5,8,7,10,9,11] => ? => ? = 12
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [11,9,10,7,5,6,3,4,8,1,2,12] => [12,2,1,8,4,3,6,5,7,10,9,11] => ? => ? = 12
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,8,5,3,4,6,1,2,12] => [12,2,1,6,4,3,5,8,7,10,9,11] => ? => ? = 12
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,5,6,8,3,4,1,2,12] => [12,2,1,4,3,8,6,5,7,10,9,11] => ? => ? = 12
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [11,9,10,7,8,5,6,3,4,1,2,12] => [12,2,1,4,3,6,5,8,7,10,9,11] => ? => ? = 12
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [11,12,9,7,8,5,3,4,1,2,6,10] => [10,6,2,1,4,3,5,8,7,9,12,11] => ? => ? = 10
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [11,12,9,7,8,5,6,3,1,2,4,10] => [10,4,2,1,3,6,5,8,7,9,12,11] => ? => ? = 10
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [11,12,9,7,8,5,3,4,6,1,2,10] => [10,2,1,6,4,3,5,8,7,9,12,11] => ? => ? = 10
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [11,12,9,7,8,5,6,3,4,1,2,10] => [10,2,1,4,3,6,5,8,7,9,12,11] => ? => ? = 10
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [11,9,10,12,7,5,6,3,1,2,4,8] => [8,4,2,1,3,6,5,7,12,10,9,11] => ? => ? = 8
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [11,12,9,10,7,5,6,3,1,2,4,8] => [8,4,2,1,3,6,5,7,10,9,12,11] => ? => ? = 8
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [11,12,9,7,8,5,6,3,4,10,1,2] => [2,1,10,4,3,6,5,8,7,9,12,11] => ? => ? = 2
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [11,12,9,7,8,10,5,3,4,1,2,6] => [6,2,1,4,3,5,10,8,7,9,12,11] => ? => ? = 6
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [11,12,9,7,8,5,6,10,3,4,1,2] => [2,1,4,3,10,6,5,8,7,9,12,11] => ? => ? = 2
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [11,12,9,7,8,5,6,10,3,1,2,4] => [4,2,1,3,10,6,5,8,7,9,12,11] => ? => ? = 4
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [11,12,9,7,8,10,5,6,3,1,2,4] => [4,2,1,3,6,5,10,8,7,9,12,11] => ? => ? = 4
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [11,12,9,7,8,10,5,3,4,6,1,2] => [2,1,6,4,3,5,10,8,7,9,12,11] => ? => ? = 2
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [11,12,9,7,8,10,5,6,3,4,1,2] => [2,1,4,3,6,5,10,8,7,9,12,11] => ? => ? = 2
[[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]],.]
=> [9,7,8,5,3,1,2,4,6,10] => ? => ? => ? = 10
[[[[.,.],[.,.]],[.,.]],[[.,.],[[.,.],.]]]
=> [9,10,7,8,5,3,1,2,4,6] => ? => ? => ? = 6
[[[[.,.],[.,.]],[.,.]],[[[.,.],[.,.]],.]]
=> [9,7,8,10,5,3,1,2,4,6] => ? => ? => ? = 6
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],.]]]]
=> [9,7,8,10,5,6,3,4,1,2] => [2,1,4,3,6,5,10,8,7,9] => ? => ? = 2
[[.,.],[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],.]]]]]
=> [11,9,10,12,7,8,5,6,3,4,1,2] => ? => ? => ? = 2
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: 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: 82% ●values known / values provided: 94%●distinct values known / distinct values provided: 82%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 94%●distinct values known / distinct values provided: 82%
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
[[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [7,5,6,3,1,2,4,8] => [5,6,4,7,2,3,1,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 8 - 1
[[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [7,5,3,4,1,2,6,8] => [5,6,3,4,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 8 - 1
[[[[.,.],[[[[.,.],.],.],.]],.],.]
=> [3,4,5,6,1,2,7,8] => [5,6,1,2,3,4,7,8] => ?
=> ? = 8 - 1
[[[[[.,.],[[[.,.],.],.]],.],.],.]
=> [3,4,5,1,2,6,7,8] => [4,5,1,2,3,6,7,8] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => [11,12,9,10,7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
=> [9,2,3,1,4,5,6,7,8] => [4,2,3,5,6,7,8,9,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8 - 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [9,7,8,5,6,3,1,2,4,10] => [7,8,6,9,4,5,2,3,1,10] => ?
=> ? = 10 - 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [9,7,8,5,3,4,1,2,6,10] => [7,8,5,6,4,9,2,3,1,10] => ?
=> ? = 10 - 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [9,7,8,5,3,4,6,1,2,10] => [8,9,5,6,4,7,2,3,1,10] => ?
=> ? = 10 - 1
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [9,7,8,5,6,3,4,1,2,10] => [8,9,6,7,4,5,2,3,1,10] => ?
=> ? = 10 - 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [9,10,7,5,6,3,1,2,4,8] => [7,8,6,9,4,5,3,10,1,2] => ?
=> ? = 8 - 1
[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [9,10,7,5,6,3,4,1,2,8] => [8,9,6,7,4,5,3,10,1,2] => ?
=> ? = 8 - 1
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [9,10,7,5,6,3,4,8,1,2] => [9,10,6,7,4,5,3,8,1,2] => ?
=> ? = 2 - 1
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [9,10,7,5,6,8,3,1,2,4] => [8,9,7,10,4,5,3,6,1,2] => ?
=> ? = 4 - 1
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [9,10,7,5,6,8,3,4,1,2] => [9,10,7,8,4,5,3,6,1,2] => ?
=> ? = 2 - 1
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [11,9,10,7,5,6,3,4,1,2,8,12] => [9,10,7,8,5,6,4,11,2,3,1,12] => ?
=> ? = 12 - 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [11,9,10,7,8,5,3,4,1,2,6,12] => [9,10,7,8,6,11,4,5,2,3,1,12] => ?
=> ? = 12 - 1
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [11,9,10,7,5,6,8,3,1,2,4,12] => [9,10,8,11,5,6,4,7,2,3,1,12] => ?
=> ? = 12 - 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,8,5,6,3,1,2,4,12] => [9,10,8,11,6,7,4,5,2,3,1,12] => ?
=> ? = 12 - 1
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [11,9,10,7,5,6,3,4,8,1,2,12] => [10,11,7,8,5,6,4,9,2,3,1,12] => ?
=> ? = 12 - 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,8,5,3,4,6,1,2,12] => [10,11,7,8,6,9,4,5,2,3,1,12] => ?
=> ? = 12 - 1
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [11,9,10,7,5,6,8,3,4,1,2,12] => [10,11,8,9,5,6,4,7,2,3,1,12] => ?
=> ? = 12 - 1
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [11,9,10,7,8,5,6,3,4,1,2,12] => [10,11,8,9,6,7,4,5,2,3,1,12] => ?
=> ? = 12 - 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [11,12,9,7,8,5,3,4,1,2,6,10] => [9,10,7,8,6,11,4,5,3,12,1,2] => ?
=> ? = 10 - 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [11,12,9,7,8,5,6,3,1,2,4,10] => [9,10,8,11,6,7,4,5,3,12,1,2] => ?
=> ? = 10 - 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [11,12,9,7,8,5,3,4,6,1,2,10] => [10,11,7,8,6,9,4,5,3,12,1,2] => ?
=> ? = 10 - 1
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [11,12,9,7,8,5,6,3,4,1,2,10] => [10,11,8,9,6,7,4,5,3,12,1,2] => ?
=> ? = 10 - 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [11,9,10,12,7,5,6,3,1,2,4,8] => [9,10,8,11,6,7,5,12,2,3,1,4] => ?
=> ? = 8 - 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [11,12,9,10,7,5,6,3,1,2,4,8] => [9,10,8,11,6,7,5,12,3,4,1,2] => ?
=> ? = 8 - 1
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [11,12,9,7,8,5,6,3,4,10,1,2] => [11,12,8,9,6,7,4,5,3,10,1,2] => ?
=> ? = 2 - 1
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [11,12,9,7,8,10,5,3,4,1,2,6] => [10,11,8,9,7,12,4,5,3,6,1,2] => ?
=> ? = 6 - 1
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [11,12,9,7,8,5,6,10,3,4,1,2] => [11,12,9,10,6,7,4,5,3,8,1,2] => ?
=> ? = 2 - 1
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [11,12,9,7,8,5,6,10,3,1,2,4] => [10,11,9,12,6,7,4,5,3,8,1,2] => ?
=> ? = 4 - 1
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [11,12,9,7,8,10,5,6,3,1,2,4] => [10,11,9,12,7,8,4,5,3,6,1,2] => ?
=> ? = 4 - 1
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [11,12,9,7,8,10,5,3,4,6,1,2] => [11,12,8,9,7,10,4,5,3,6,1,2] => ?
=> ? = 2 - 1
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [11,12,9,7,8,10,5,6,3,4,1,2] => [11,12,9,10,7,8,4,5,3,6,1,2] => ?
=> ? = 2 - 1
[[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]],.]
=> [9,7,8,5,3,1,2,4,6,10] => ? => ?
=> ? = 10 - 1
[[[[.,.],[.,.]],[.,.]],[[.,.],[[.,.],.]]]
=> [9,10,7,8,5,3,1,2,4,6] => [7,8,6,9,5,10,3,4,1,2] => ?
=> ? = 6 - 1
[[[[.,.],[.,.]],[.,.]],[[[.,.],[.,.]],.]]
=> [9,7,8,10,5,3,1,2,4,6] => ? => ?
=> ? = 6 - 1
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],.]]]]
=> [9,7,8,10,5,6,3,4,1,2] => ? => ?
=> ? = 2 - 1
[[.,.],[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],.]]]]]
=> [11,9,10,12,7,8,5,6,3,4,1,2] => ? => ?
=> ? = 2 - 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: 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: 82% ●values known / values provided: 91%●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: 82% ●values known / values provided: 91%●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
[[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,2] => [4,2,2] => ? = 2
[[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,6] => [6,2] => ? = 2
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,6] => [6,2] => ? = 2
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => [6,2] => ? = 2
[[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => [2,2,4] => ? = 4
[[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [6,2] => [2,6] => ? = 6
[[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [6,2] => [2,6] => ? = 6
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => [2,6] => ? = 6
[[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[.,.],[[[[[.,.],.],.],.],.]],.]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [8] => [8] => ? = 8
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[.,.],[[[[.,.],.],.],.]],.],.]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[[.,.],[[[.,.],.],.]],.],.],.]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[[.,[[[.,.],.],.]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[[[.,.],[[.,.],.]],.],.],.],.]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2,2] => [2,2,2,2,2] => ? = 2
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [10] => [10] => ? = 10
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [10] => [10] => ? = 10
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [10] => [10] => ? = 10
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [10] => [10] => ? = 10
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [8,2] => [2,8] => ? = 8
[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [8,2] => [2,8] => ? = 8
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,2] => [6,2,2] => ? = 2
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2] => [4,2,4] => ? = 4
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,2,4,2] => [2,4,2,2] => ? = 2
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [12] => [12] => ? = 12
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [12] => [12] => ? = 12
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [12] => [12] => ? = 12
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [12] => [12] => ? = 12
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [12] => [12] => ? = 12
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [12] => [12] => ? = 12
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [12] => [12] => ? = 12
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [12] => [12] => ? = 12
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [10,2] => [2,10] => ? = 10
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [10,2] => [2,10] => ? = 10
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [10,2] => [2,10] => ? = 10
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [10,2] => [2,10] => ? = 10
Description
The last part of an integer composition.
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: 73% ●values known / values provided: 91%●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: 73% ●values known / values provided: 91%●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
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => [8,7,1,2,3,4,5,6] => [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
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [3,4,5,6,7,2,8,1] => [8,6,1,2,3,4,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> [3,4,5,6,2,7,8,1] => [8,5,1,2,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [2,3,5,4,6,7,8,1] => [8,1,2,4,3,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [8,3,2,1,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [2,4,3,5,6,7,8,1] => [8,1,3,2,4,5,6,7] => [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
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [2,3,4,1,5,6,8,7] => [4,1,2,3,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [3,2,1,4,5,6,8,7] => [3,2,1,4,5,6,8,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [2,3,1,4,5,6,8,7] => [3,1,2,4,5,6,8,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> [3,4,5,6,7,2,1,8] => [7,6,1,2,3,4,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [3,4,5,6,2,7,1,8] => [7,5,1,2,3,4,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [2,3,4,5,6,7,1,8] => [7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [2,3,4,5,6,1,7,8] => [6,1,2,3,4,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8 - 1
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [2,3,4,5,1,6,7,8] => [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[[[[.,[[[.,.],.],.]],.],.],.],.]
=> [2,3,4,1,5,6,7,8] => [4,1,2,3,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [2,3,1,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 - 1
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,1] => [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,3,5,7,9,10,8,6,4,2] => [1,10,2,9,3,8,4,7,5,6] => ?
=> ? = 2 - 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]]
=> [1,3,5,7,9,11,12,10,8,6,4,2] => [1,12,2,11,3,10,4,9,5,8,6,7] => ?
=> ? = 2 - 1
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8 - 1
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9 - 1
[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
=> [1,3,2,4,5,6,7,9,8] => [1,3,2,4,5,6,7,9,8] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8 - 1
[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
=> [2,3,1,4,5,6,7,9,8] => [3,1,2,4,5,6,7,9,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8 - 1
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [2,4,3,5,6,7,8,9,1] => [9,1,3,2,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> ? => ? => ?
=> ? = 10 - 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> ? => ? => ?
=> ? = 10 - 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> ? => ? => ?
=> ? = 10 - 1
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> ? => ? => ?
=> ? = 10 - 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> ? => ? => ?
=> ? = 8 - 1
[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> ? => ? => ?
=> ? = 8 - 1
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> ? => ? => ?
=> ? = 2 - 1
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> ? => ? => ?
=> ? = 4 - 1
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> ? => ? => ?
=> ? = 2 - 1
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> ? => ? => ?
=> ? = 12 - 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> ? => ? => ?
=> ? = 12 - 1
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> ? => ? => ?
=> ? = 12 - 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> ? => ? => ?
=> ? = 12 - 1
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> ? => ? => ?
=> ? = 12 - 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> ? => ? => ?
=> ? = 12 - 1
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> ? => ? => ?
=> ? = 12 - 1
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> ? => ? => ?
=> ? = 12 - 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> ? => ? => ?
=> ? = 10 - 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> ? => ? => ?
=> ? = 10 - 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: 64% ●values known / values provided: 87%●distinct values known / distinct values provided: 64%
St000026: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 87%●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
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 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
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2
[[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4
[[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 6
[[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 6
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 7
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 7
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 7
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
[[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 8
[[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 8
[[[.,.],[[[[[.,.],.],.],.],.]],.]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 8
[[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 8
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 8
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 8
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
[[[[.,.],[[[[.,.],.],.],.]],.],.]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
[[[[[.,.],[[[.,.],.],.]],.],.],.]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 8
[[[[[.,[[[.,.],.],.]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[.,.],[[.,.],.]],.],.],.],.]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 8
[[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8
[[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
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: 64% ●values known / values provided: 87%●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: 64% ●values known / values provided: 87%●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
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => [1,2,8,7,6,5,4,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 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
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [3,4,5,6,7,2,8,1] => [1,8,2,7,6,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> [3,4,5,6,2,7,8,1] => [1,8,7,2,6,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [2,3,5,4,6,7,8,1] => [1,8,7,6,4,5,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [2,4,3,5,6,7,8,1] => [1,8,7,6,5,3,4,2] => [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
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,3,5,7,8,6,4,2] => [2,4,6,8,7,5,3,1] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,3,5,7,6,8,4,2] => [2,4,8,6,7,5,3,1] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,3,5,4,7,8,6,2] => [2,6,8,7,4,5,3,1] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
[[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,3,5,7,6,4,8,2] => [2,8,4,6,7,5,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [1,3,5,4,7,6,8,2] => [2,8,6,7,4,5,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,3,4,5,6,7,8,2] => [2,8,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,3,2,5,7,8,6,4] => [4,6,8,7,5,2,3,1] => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4
[[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> [1,3,2,5,7,6,8,4] => [4,8,6,7,5,2,3,1] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
[[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [1,3,5,4,2,7,8,6] => [6,8,7,2,4,5,3,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6
[[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,3,2,5,4,7,8,6] => [6,8,7,4,5,2,3,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,2,3,4,5,7,8,6] => [6,8,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [2,3,4,1,5,6,8,7] => [7,8,6,5,1,4,3,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,2,4,3,5,6,8,7] => [7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [3,2,1,4,5,6,8,7] => [7,8,6,5,4,1,2,3] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [2,3,1,4,5,6,8,7] => [7,8,6,5,4,1,3,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,3,2,4,5,6,8,7] => [7,8,6,5,4,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,8,7] => [7,8,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,8,7] => [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> [3,4,5,6,7,2,1,8] => [8,1,2,7,6,5,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [3,4,5,6,2,7,1,8] => [8,1,7,2,6,5,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [2,3,4,5,6,7,1,8] => [8,1,7,6,5,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> [1,3,5,7,6,4,2,8] => [8,2,4,6,7,5,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [1,3,5,4,7,6,2,8] => [8,2,6,7,4,5,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[.,.],[[[[[.,.],.],.],.],.]],.]
=> [1,3,4,5,6,7,2,8] => [8,2,7,6,5,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,3,2,5,7,6,4,8] => [8,4,6,7,5,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [1,3,5,4,2,7,6,8] => [8,6,7,2,4,5,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => [8,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => [8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [2,3,4,5,6,1,7,8] => [8,7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[.,.],[[[[.,.],.],.],.]],.],.]
=> [1,3,4,5,6,2,7,8] => [8,7,2,6,5,4,3,1] => ?
=> ? = 8
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [2,3,4,5,1,6,7,8] => [8,7,6,1,5,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[.,.],[[[.,.],.],.]],.],.],.]
=> [1,3,4,5,2,6,7,8] => [8,7,6,2,5,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[.,[[[.,.],.],.]],.],.],.],.]
=> [2,3,4,1,5,6,7,8] => [8,7,6,5,1,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[.,.],[[.,.],.]],.],.],.],.]
=> [1,3,4,2,5,6,7,8] => [8,7,6,5,2,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [2,3,1,4,5,6,7,8] => [8,7,6,5,4,1,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,1] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
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: St000987
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 87%●distinct values known / distinct values provided: 64%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 87%●distinct values known / distinct values provided: 64%
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
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 1 - 1
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7] => ([(6,7)],8)
=> ? = 2 - 1
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,7] => ([(6,7)],8)
=> ? = 2 - 1
[[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,7] => ([(6,7)],8)
=> ? = 2 - 1
[[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,7] => ([(6,7)],8)
=> ? = 2 - 1
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,7] => ([(6,7)],8)
=> ? = 2 - 1
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,7] => ([(6,7)],8)
=> ? = 2 - 1
[[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[.,.],[[[[[.,.],.],.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[.,.],[[[[.,.],.],.],.]],.],.]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[.,.],[[[.,.],.],.]],.],.],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[.,[[[.,.],.],.]],.],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[[.,.],[[.,.],.]],.],.],.],.]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9] => ([],9)
=> ? = 1 - 1
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
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. St001725The harmonious chromatic number of a graph. St000054The first entry of the permutation. St000740The last entry of a permutation. 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. 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.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!