Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000975
(load all 3 compositions to match this statistic)
Mapping
St000975: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => 1
[[],[]]
 => 2
[[[]]]
 => 2
[[],[],[]]
 => 2
[[],[[]]]
 => 3
[[[]],[]]
 => 3
[[[],[]]]
 => 3
[[[[]]]]
 => 3
[[],[],[],[]]
 => 2
[[],[],[[]]]
 => 3
[[],[[]],[]]
 => 2
[[],[[],[]]]
 => 3
[[],[[[]]]]
 => 4
[[[]],[],[]]
 => 3
[[[]],[[]]]
 => 4
[[[],[]],[]]
 => 3
[[[[]]],[]]
 => 4
[[[],[],[]]]
 => 3
[[[],[[]]]]
 => 4
[[[[]],[]]]
 => 4
[[[[],[]]]]
 => 4
[[[[[]]]]]
 => 4
[[],[],[],[],[]]
 => 2
[[],[],[],[[]]]
 => 3
[[],[],[[]],[]]
 => 2
[[],[],[[],[]]]
 => 3
[[],[],[[[]]]]
 => 4
[[],[[]],[],[]]
 => 2
[[],[[]],[[]]]
 => 3
[[],[[],[]],[]]
 => 2
[[],[[[]]],[]]
 => 2
[[],[[],[],[]]]
 => 3
[[],[[],[[]]]]
 => 4
[[],[[[]],[]]]
 => 3
[[],[[[],[]]]]
 => 4
[[],[[[[]]]]]
 => 5
[[[]],[],[],[]]
 => 3
[[[]],[],[[]]]
 => 4
[[[]],[[]],[]]
 => 3
[[[]],[[],[]]]
 => 4
[[[]],[[[]]]]
 => 5
[[[],[]],[],[]]
 => 3
[[[[]]],[],[]]
 => 4
[[[],[]],[[]]]
 => 4
[[[[]]],[[]]]
 => 5
[[[],[],[]],[]]
 => 3
[[[],[[]]],[]]
 => 3
[[[[]],[]],[]]
 => 4
[[[[],[]]],[]]
 => 4
[[[[[]]]],[]]
 => 5
Description
The length of the boundary minus the length of the trunk of an ordered tree. This is the size of the set of edges which are either on the left most path or on the right most path from the root.
Matching statistic: St001004
(load all 8 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => [1] => 1
[[],[]]
 => [[.,.],.]
 => [1,2] => [2,1] => 2
[[[]]]
 => [.,[.,.]]
 => [2,1] => [1,2] => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => [3,2,1] => 2
[[],[[]]]
 => [[.,.],[.,.]]
 => [1,3,2] => [2,3,1] => 3
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => 3
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [1,3,2] => 3
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => 3
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => 2
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [3,4,2,1] => 3
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [4,2,3,1] => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [2,4,3,1] => 3
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,3,4,1] => 4
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => 3
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [3,4,1,2] => 4
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,1,3,2] => 3
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => 4
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,4,3,2] => 3
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,3,4,2] => 4
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,4,2,3] => 4
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,4,3] => 4
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => 4
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [5,4,3,2,1] => 2
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [4,5,3,2,1] => 3
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [5,3,4,2,1] => 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [3,5,4,2,1] => 3
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [3,4,5,2,1] => 4
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => [5,4,2,3,1] => 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [4,5,2,3,1] => 3
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [5,2,4,3,1] => 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [5,2,3,4,1] => 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [2,5,4,3,1] => 3
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [2,4,5,3,1] => 4
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [2,5,3,4,1] => 3
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [2,3,5,4,1] => 4
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,3,4,5,1] => 5
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [5,4,3,1,2] => 3
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [4,5,3,1,2] => 4
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [5,3,4,1,2] => 3
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,5,4,1,2] => 4
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [3,4,5,1,2] => 5
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [5,4,1,3,2] => 3
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [5,4,1,2,3] => 4
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [4,5,1,3,2] => 4
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [4,5,1,2,3] => 5
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [5,1,4,3,2] => 3
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [5,1,3,4,2] => 3
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [5,1,4,2,3] => 4
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [5,1,2,4,3] => 4
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [5,1,2,3,4] => 5
[[],[],[],[],[],[[]]]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 3
[[],[],[],[],[[]],[]]
 => [[[[[[.,.],.],.],.],[.,.]],.]
 => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[[],[],[],[],[[],[]]]
 => [[[[[.,.],.],.],.],[[.,.],.]]
 => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 3
[[],[],[],[],[[[]]]]
 => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => ? = 4
[[],[],[],[[]],[],[]]
 => [[[[[[.,.],.],.],[.,.]],.],.]
 => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[[],[],[],[[]],[[]]]
 => [[[[[.,.],.],.],[.,.]],[.,.]]
 => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 3
[[],[],[],[[],[]],[]]
 => [[[[[.,.],.],.],[[.,.],.]],.]
 => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 2
[[],[],[],[[[]]],[]]
 => [[[[[.,.],.],.],[.,[.,.]]],.]
 => [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => ? = 2
[[],[],[],[[],[],[]]]
 => [[[[.,.],.],.],[[[.,.],.],.]]
 => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 3
[[],[],[],[[],[[]]]]
 => [[[[.,.],.],.],[[.,.],[.,.]]]
 => [1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => ? = 4
[[],[],[],[[[]],[]]]
 => [[[[.,.],.],.],[[.,[.,.]],.]]
 => [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => ? = 3
[[],[],[],[[[],[]]]]
 => [[[[.,.],.],.],[.,[[.,.],.]]]
 => [1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => ? = 4
[[],[],[],[[[[]]]]]
 => [[[[.,.],.],.],[.,[.,[.,.]]]]
 => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ? = 5
[[],[],[[]],[],[],[]]
 => [[[[[[.,.],.],[.,.]],.],.],.]
 => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 2
[[],[],[[]],[],[[]]]
 => [[[[[.,.],.],[.,.]],.],[.,.]]
 => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 3
[[],[],[[]],[[]],[]]
 => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 2
[[],[],[[]],[[],[]]]
 => [[[[.,.],.],[.,.]],[[.,.],.]]
 => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => ? = 3
[[],[],[[]],[[[]]]]
 => [[[[.,.],.],[.,.]],[.,[.,.]]]
 => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => ? = 4
[[],[],[[],[]],[],[]]
 => [[[[[.,.],.],[[.,.],.]],.],.]
 => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => ? = 2
[[],[],[[[]]],[],[]]
 => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 2
[[],[],[[],[]],[[]]]
 => [[[[.,.],.],[[.,.],.]],[.,.]]
 => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => ? = 3
[[],[],[[[]]],[[]]]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => ? = 3
[[],[],[[],[],[]],[]]
 => [[[[.,.],.],[[[.,.],.],.]],.]
 => [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => ? = 2
[[],[],[[],[[]]],[]]
 => [[[[.,.],.],[[.,.],[.,.]]],.]
 => [1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => ? = 2
[[],[],[[[]],[]],[]]
 => [[[[.,.],.],[[.,[.,.]],.]],.]
 => [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => ? = 2
[[],[],[[[],[]]],[]]
 => [[[[.,.],.],[.,[[.,.],.]]],.]
 => [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => ? = 2
[[],[],[[[[]]]],[]]
 => [[[[.,.],.],[.,[.,[.,.]]]],.]
 => [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => ? = 2
[[],[],[[],[],[],[]]]
 => [[[.,.],.],[[[[.,.],.],.],.]]
 => [1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => ? = 3
[[],[],[[],[],[[]]]]
 => [[[.,.],.],[[[.,.],.],[.,.]]]
 => [1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => ? = 4
[[],[],[[],[[]],[]]]
 => [[[.,.],.],[[[.,.],[.,.]],.]]
 => [1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => ? = 3
[[],[],[[],[[],[]]]]
 => [[[.,.],.],[[.,.],[[.,.],.]]]
 => [1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => ? = 4
[[],[],[[],[[[]]]]]
 => [[[.,.],.],[[.,.],[.,[.,.]]]]
 => [1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => ? = 5
[[],[],[[[]],[],[]]]
 => [[[.,.],.],[[[.,[.,.]],.],.]]
 => [1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => ? = 3
[[],[],[[[]],[[]]]]
 => [[[.,.],.],[[.,[.,.]],[.,.]]]
 => [1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => ? = 4
[[],[],[[[],[]],[]]]
 => [[[.,.],.],[[.,[[.,.],.]],.]]
 => [1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => ? = 3
[[],[],[[[[]]],[]]]
 => [[[.,.],.],[[.,[.,[.,.]]],.]]
 => [1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => ? = 3
[[],[],[[[],[],[]]]]
 => [[[.,.],.],[.,[[[.,.],.],.]]]
 => [1,2,5,6,7,4,3] => [3,4,7,6,5,2,1] => ? = 4
[[],[],[[[],[[]]]]]
 => [[[.,.],.],[.,[[.,.],[.,.]]]]
 => [1,2,5,7,6,4,3] => [3,4,6,7,5,2,1] => ? = 5
[[],[],[[[[]],[]]]]
 => [[[.,.],.],[.,[[.,[.,.]],.]]]
 => [1,2,6,5,7,4,3] => [3,4,7,5,6,2,1] => ? = 4
[[],[],[[[[],[]]]]]
 => [[[.,.],.],[.,[.,[[.,.],.]]]]
 => [1,2,6,7,5,4,3] => [3,4,5,7,6,2,1] => ? = 5
[[],[],[[[[[]]]]]]
 => [[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => ? = 6
[[],[[]],[],[],[],[]]
 => [[[[[[.,.],[.,.]],.],.],.],.]
 => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 2
[[],[[]],[],[],[[]]]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => ? = 3
[[],[[]],[],[[]],[]]
 => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 2
[[],[[]],[],[[],[]]]
 => [[[[.,.],[.,.]],.],[[.,.],.]]
 => [1,3,2,4,6,7,5] => [5,7,6,4,2,3,1] => ? = 3
[[],[[]],[],[[[]]]]
 => [[[[.,.],[.,.]],.],[.,[.,.]]]
 => [1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => ? = 4
[[],[[]],[[]],[],[]]
 => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => ? = 2
[[],[[]],[[]],[[]]]
 => [[[[.,.],[.,.]],[.,.]],[.,.]]
 => [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ? = 3
[[],[[]],[[],[]],[]]
 => [[[[.,.],[.,.]],[[.,.],.]],.]
 => [1,3,2,5,6,4,7] => [7,4,6,5,2,3,1] => ? = 2
[[],[[]],[[[]]],[]]
 => [[[[.,.],[.,.]],[.,[.,.]]],.]
 => [1,3,2,6,5,4,7] => [7,4,5,6,2,3,1] => ? = 2
Description
The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].