Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St001004
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => [1] => 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => [1,2] => 2
[[[]]]
 => [[.,.],.]
 => [1,2] => [2,1] => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => 3
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => 3
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => 3
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => [3,2,1] => 2
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => [2,3,1] => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => 4
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,2,3] => 4
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,4,2,3] => 4
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,1,2] => 3
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,4,1,2] => 4
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,4,3] => 4
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,3,2] => 3
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,4,3,2] => 3
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,3,4,2] => 4
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,3,2,1] => 2
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,4,2,1] => 3
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,3,1] => 2
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [3,2,4,1] => 3
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [2,3,4,1] => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 5
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,3,4] => 5
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,5,2,3,4] => 5
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,1,2,3] => 4
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,5,1,2,3] => 5
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,5,3,4] => 5
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,1,4,2,3] => 4
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,5,4,2,3] => 4
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,5,2,3] => 5
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,3,1,2] => 3
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,5,3,1,2] => 4
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,3,4,1,2] => 3
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,3,5,1,2] => 4
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,4,5,1,2] => 5
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,5,4] => 5
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,1,2,4,3] => 4
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,5,2,4,3] => 4
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,4,1,3,2] => 3
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,5,1,3,2] => 4
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,2,5,4,3] => 4
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,4,5,3] => 5
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,1,4,3,2] => 3
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,3,4,2] => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,5,4,3,2] => 3
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,5,3,2] => 4
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,5,3,4,2] => 3
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,4,3,5,2] => 4
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,3,4,5,2] => 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]].