Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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: 100% ●values known / values provided: 100%●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
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]].
Matching statistic: St000975
(load all 3 compositions to match this statistic)
Mapping
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
St000975: Ordered trees ⟶ ℤResult quality: 86% ●values known / values provided: 86%●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
[[],[],[],[],[[[]],[]]]
 => [[[],[[]]],[],[],[],[]]
 => ? = 3
[[],[],[],[[[]],[],[]]]
 => [[[],[],[[]]],[],[],[]]
 => ? = 3
[[],[],[[[]]],[[[]]]]
 => [[[[]]],[[[]]],[],[]]
 => ? = 4
[[],[],[[],[[],[],[]]]]
 => [[[[],[],[]],[]],[],[]]
 => ? = 4
[[],[],[[[]],[],[],[]]]
 => [[[],[],[],[[]]],[],[]]
 => ? = 3
[[],[],[[[],[],[]],[]]]
 => [[[],[[],[],[]]],[],[]]
 => ? = 3
[[],[],[[[[[]]],[]]]]
 => [[[[],[[[]]]]],[],[]]
 => ? = 4
[[],[[[]]],[],[[[]]]]
 => [[[[]]],[],[[[]]],[]]
 => ? = 4
[[],[[],[],[],[[],[]]]]
 => [[[[],[]],[],[],[]],[]]
 => ? = 4
[[],[[],[],[[],[]],[]]]
 => [[[],[[],[]],[],[]],[]]
 => ? = 3
[[],[[[]],[],[],[],[]]]
 => [[[],[],[],[],[[]]],[]]
 => ? = 3
[[],[[[],[[[],[]]]]]]
 => [[[[[[],[]]],[]]],[]]
 => ? = 6
[[],[[[[[]]],[],[]]]]
 => [[[[],[],[[[]]]]],[]]
 => ? = 4
[[],[[[[[[]]],[]]]]]
 => [[[[[],[[[]]]]]],[]]
 => ? = 5
[[],[[[[[[[]]]]]]]]
 => [[[[[[[[]]]]]]],[]]
 => ? = 8
[[[]],[[]],[[],[[]]]]
 => [[[[]],[]],[[]],[[]]]
 => ? = 5
[[[]],[[]],[[[[]]]]]
 => [[[[[]]]],[[]],[[]]]
 => ? = 6
[[[]],[[[[[[]]]]]]]
 => [[[[[[[]]]]]],[[]]]
 => ? = 8
[[[],[],[]],[[],[],[]]]
 => [[[],[],[]],[[],[],[]]]
 => ? = 4
[[[],[[]]],[[[]],[]]]
 => [[[],[[]]],[[[]],[]]]
 => ? = 4
[[[[]],[]],[[],[[]]]]
 => [[[[]],[]],[[],[[]]]]
 => ? = 6
[[[[],[]]],[[[],[]]]]
 => [[[[],[]]],[[[],[]]]]
 => ? = 6
[[[[[]]]],[[[[]]]]]
 => [[[[[]]]],[[[[]]]]]
 => ? = 8
[[[[[[]]]]],[[[]]]]
 => [[[[]]],[[[[[]]]]]]
 => ? = 8
[[[],[],[],[],[],[],[]]]
 => [[[],[],[],[],[],[],[]]]
 => ? = 3
[[[],[],[],[],[],[[]]]]
 => [[[[]],[],[],[],[],[]]]
 => ? = 4
[[[],[],[],[],[[]],[]]]
 => [[[],[[]],[],[],[],[]]]
 => ? = 3
[[[],[],[],[],[[],[]]]]
 => [[[[],[]],[],[],[],[]]]
 => ? = 4
[[[],[],[],[],[[[]]]]]
 => [[[[[]]],[],[],[],[]]]
 => ? = 5
[[[],[],[],[[]],[],[]]]
 => [[[],[],[[]],[],[],[]]]
 => ? = 3
[[[],[],[],[[],[],[]]]]
 => [[[[],[],[]],[],[],[]]]
 => ? = 4
[[[],[],[[]],[[]],[]]]
 => [[[],[[]],[[]],[],[]]]
 => ? = 3
[[[],[],[[],[]],[],[]]]
 => [[[],[],[[],[]],[],[]]]
 => ? = 3
[[[],[],[[[]]],[],[]]]
 => [[[],[],[[[]]],[],[]]]
 => ? = 3
[[[],[],[[],[],[]],[]]]
 => [[[],[[],[],[]],[],[]]]
 => ? = 3
[[[],[[]],[],[],[[]]]]
 => [[[[]],[],[],[[]],[]]]
 => ? = 4
[[[],[[],[]],[[],[]]]]
 => [[[[],[]],[[],[]],[]]]
 => ? = 4
[[[],[[[]]],[[[]]]]]
 => [[[[[]]],[[[]]],[]]]
 => ? = 5
[[[],[[],[[],[[]]]]]]
 => [[[[[[]],[]],[]],[]]]
 => ? = 6
[[[[]],[],[],[],[],[]]]
 => [[[],[],[],[],[],[[]]]]
 => ? = 4
[[[[]],[[],[[]],[]]]]
 => [[[[],[[]],[]],[[]]]]
 => ? = 5
[[[[[[]]]],[[[]]]]]
 => [[[[[]]],[[[[]]]]]]
 => ? = 8
[[[[[[[]]]]],[[]]]]
 => [[[[]],[[[[[]]]]]]]
 => ? = 8
[[[[[[]],[]],[]],[]]]
 => [[[],[[],[[],[[]]]]]]
 => ? = 6
[[[[[[[]]]],[]],[]]]
 => [[[],[[],[[[[]]]]]]]
 => ? = 7
[[[[[[[]],[]]]],[]]]
 => [[[],[[[[],[[]]]]]]]
 => ? = 7
[[[[[[[[]]]]]],[]]]
 => [[[],[[[[[[]]]]]]]]
 => ? = 8
[[[[],[],[],[[[]]]]]]
 => [[[[[[]]],[],[],[]]]]
 => ? = 6
[[[[],[[[]]],[],[]]]]
 => [[[[],[],[[[]]],[]]]]
 => ? = 4
[[[[[[]]],[],[],[]]]]
 => [[[[],[],[],[[[]]]]]]
 => ? = 6
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.