Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000051
Mapping
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [.,.]
 => 0
[.,[.,.]]
 => [.,[.,.]]
 => 0
[[.,.],.]
 => [[.,.],.]
 => 1
[.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => 0
[.,[[.,.],.]]
 => [.,[[.,.],.]]
 => 0
[[.,.],[.,.]]
 => [[.,[.,.]],.]
 => 2
[[.,[.,.]],.]
 => [[.,.],[.,.]]
 => 1
[[[.,.],.],.]
 => [[[.,.],.],.]
 => 2
[.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => 0
[.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => 0
[.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => 0
[.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => 0
[.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => 0
[[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => 3
[[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => 3
[[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => 2
[[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => 3
[[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => 1
[[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => 1
[[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => 3
[[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => 2
[[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => 3
[.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => 0
[.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => 0
[.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => 0
[.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => 0
[.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => 0
[.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => 0
[.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => 0
[.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => 4
[[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => 4
[[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => 4
[[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => 4
[[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 3
[[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => 4
[[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => 4
[[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => 4
[[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => 3
[[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => 4
Description
The size of the left subtree of a binary tree.
Matching statistic: St000133
(load all 6 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => 0
[[.,.],.]
 => [1,2] => [1,2] => 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
 => [1,2,3] => [1,3,2] => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,4,3,1] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,4,3,1] => 0
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,3,2] => 3
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => 3
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,4,3,2] => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,4,1,3] => 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,4,3,2] => 3
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,4,3] => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,4,3,2] => 3
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [3,5,4,2,1] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,5,4,2,1] => 0
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [2,5,4,3,1] => 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [2,5,4,3,1] => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,5,2,4,1] => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [2,5,4,3,1] => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,5,4,1] => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,5,4,3,1] => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,4,3,2] => 4
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,4,3,2] => 4
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,4,3,2] => 4
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => 4
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,4,3,2] => 4
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,4,3] => 3
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,5,4,3,2] => 4
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,5,4,3,2] => 4
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,1,5,4] => 2
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,5,1,4,3] => 2
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,5,4,3,2] => 4
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,5,4,3] => 3
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,5,4,3,2] => 4
Description
The "bounce" of a permutation.
Matching statistic: St000026
(load all 3 compositions to match this statistic)
Mapping
Mp00018: Binary trees —left border symmetry⟶ Binary trees
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 = 0 + 1
[.,[.,.]]
 => [.,[.,.]]
 => [1,0,1,0]
 => 1 = 0 + 1
[[.,.],.]
 => [[.,.],.]
 => [1,1,0,0]
 => 2 = 1 + 1
[.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,.],.]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,.]]
 => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[[.,[.,.]],.]
 => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],.],.]
 => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000740
(load all 17 compositions to match this statistic)
Mapping
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [.,.]
 => [1] => 1 = 0 + 1
[.,[.,.]]
 => [.,[.,.]]
 => [2,1] => 1 = 0 + 1
[[.,.],.]
 => [[.,.],.]
 => [1,2] => 2 = 1 + 1
[.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 1 = 0 + 1
[.,[[.,.],.]]
 => [.,[[.,.],.]]
 => [2,3,1] => 1 = 0 + 1
[[.,.],[.,.]]
 => [[.,[.,.]],.]
 => [2,1,3] => 3 = 2 + 1
[[.,[.,.]],.]
 => [[.,.],[.,.]]
 => [3,1,2] => 2 = 1 + 1
[[[.,.],.],.]
 => [[[.,.],.],.]
 => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 4 = 3 + 1
[[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 4 = 3 + 1
[[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 4 = 3 + 1
[[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 3 = 2 + 1
[[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => 5 = 4 + 1
[[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 5 = 4 + 1
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St001225
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1,0]
 => 0
[.,[.,.]]
 => [2,1] => [2,1] => [1,1,0,0]
 => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[[.,.],[.,.]]
 => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 3
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 3
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 4
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 4
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 4
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 3
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
Description
The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001227
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1,0]
 => 0
[.,[.,.]]
 => [2,1] => [2,1] => [1,1,0,0]
 => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 4
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 4
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 4
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000007
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 1 = 0 + 1
[.,[.,.]]
 => [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[[.,.],.]
 => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => [1,2,3] => 1 = 0 + 1
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[[[.,.],.],.]
 => [1,2,3] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => [3,1,2,4] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,4,3,1] => [1,3,2,4] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => [2,1,3,4] => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,4,3,1] => [1,3,2,4] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,4,1,3] => [1,4,2,3] => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,4,3] => [1,4,3,2] => 3 = 2 + 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => [4,3,1,2,5] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [3,5,4,2,1] => [4,1,3,2,5] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => [4,2,1,3,5] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,5,4,2,1] => [4,1,3,2,5] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,5] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [2,5,4,3,1] => [1,4,3,2,5] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,5] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [2,5,4,3,1] => [1,4,3,2,5] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => [3,2,1,4,5] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,5,2,4,1] => [3,1,4,2,5] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [2,5,4,3,1] => [1,4,3,2,5] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,5,4,1] => [2,1,4,3,5] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,5,4,3,1] => [1,4,3,2,5] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [1,5,4,3,2] => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,3] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,5,1,4,3] => [1,5,4,2,3] => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,5,4,3] => [1,5,4,3,2] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 5 = 4 + 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000054
(load all 6 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 1 = 0 + 1
[.,[.,.]]
 => [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[[.,.],.]
 => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [1,3,2] => 1 = 0 + 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [3,1,2] => 3 = 2 + 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [1,4,3,2] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [1,3,4,2] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 4 = 3 + 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [2,4,3,1] => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 4 = 3 + 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3 = 2 + 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [1,5,4,3,2] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [1,5,4,2,3] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [1,5,3,4,2] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [1,5,2,3,4] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [1,5,3,2,4] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [1,4,5,3,2] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [1,4,5,2,3] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [1,3,4,5,2] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [1,4,3,5,2] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [1,2,4,3,5] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [1,3,4,2,5] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [5,1,4,3,2] => 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [5,1,3,4,2] => 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [4,5,1,3,2] => 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 5 = 4 + 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Matching statistic: St000382
Mapping
Mp00018: Binary trees —left border symmetry⟶ Binary trees
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 = 0 + 1
[.,[.,.]]
 => [.,[.,.]]
 => [1,0,1,0]
 => [1,1] => 1 = 0 + 1
[[.,.],.]
 => [[.,.],.]
 => [1,1,0,0]
 => [2] => 2 = 1 + 1
[.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1 = 0 + 1
[.,[[.,.],.]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,2] => 1 = 0 + 1
[[.,.],[.,.]]
 => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [3] => 3 = 2 + 1
[[.,[.,.]],.]
 => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [2,1] => 2 = 1 + 1
[[[.,.],.],.]
 => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [4] => 4 = 3 + 1
[[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [4] => 4 = 3 + 1
[[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [4] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [4] => 4 = 3 + 1
[[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 3 = 2 + 1
[[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => 5 = 4 + 1
[[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => 5 = 4 + 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ 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 = 0 + 1
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1] => 1 = 0 + 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2] => 2 = 1 + 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1 = 0 + 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1] => 1 = 0 + 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [3] => 3 = 2 + 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,2] => 2 = 1 + 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => 4 = 3 + 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [4] => 4 = 3 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => 4 = 3 + 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3 = 2 + 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => 5 = 4 + 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => 5 = 4 + 1
Description
The last part of an integer composition.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix.