Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000994
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => [1,3,2] => 1
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => [1,3,2] => 1
[[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => [1,2,3] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [1,2,3] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => [1,4,2,3] => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,1,2] => [1,4,2,3] => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [4,1,3,2] => [1,4,2,3] => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,2,1,3] => [1,4,3,2] => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,4,3,2] => [1,2,4,3] => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,4,2,3] => [1,2,4,3] => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,1,4,2] => [1,3,4,2] => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,4,3] => [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,2,4,3] => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,4,3,1,2] => [1,5,2,4,3] => 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [5,4,1,3,2] => [1,5,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,2,1,3] => [1,5,3,2,4] => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,4,1,2,3] => [1,5,3,2,4] => 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [5,1,4,3,2] => [1,5,2,3,4] => 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,1,4,2,3] => [1,5,3,4,2] => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [5,3,1,4,2] => [1,5,2,3,4] => 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [5,1,2,4,3] => [1,5,3,2,4] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,3,1,2,4] => [1,5,4,2,3] => 2
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,1,3,2,4] => [1,5,4,2,3] => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,2,1,3,4] => [1,5,4,3,2] => 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,2,5,3,4] => 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,5,4,2,3] => [1,2,5,3,4] => 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,5,2,4,3] => [1,2,5,3,4] => 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,5,3,2,4] => [1,2,5,4,3] => 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,5,2,3,4] => [1,2,5,4,3] => 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,1,5,3,2] => [1,4,3,5,2] => 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,1,5,2,3] => [1,4,2,3,5] => 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,5,3,4] => [1,2,3,5,4] => 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,3,1,5,2] => [1,4,5,2,3] => 2
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [4,1,2,5,3] => [1,4,5,3,2] => 2
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,4,2,5,3] => [1,2,4,5,3] => 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [3,1,2,5,4] => [1,3,2,4,5] => 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
Description
The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$. Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000628
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[.,.]
 => [1,0]
 => 10 => 10 => 1 = 0 + 1
[.,[.,.]]
 => [1,0,1,0]
 => 1010 => 0110 => 1 = 0 + 1
[[.,.],.]
 => [1,1,0,0]
 => 1100 => 1010 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 101010 => 100110 => 2 = 1 + 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 101100 => 110010 => 2 = 1 + 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 110010 => 010110 => 1 = 0 + 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 110100 => 011010 => 1 = 0 + 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 111000 => 101010 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 01100110 => 2 = 1 + 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 01110010 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 10100110 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 10110010 => 2 = 1 + 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 11010010 => 2 = 1 + 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 00110110 => 2 = 1 + 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 00111010 => 2 = 1 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 10010110 => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 01010110 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 10011010 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 11001010 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 01011010 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 01101010 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 10101010 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? => ? = 1 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? => ? = 1 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? => ? = 1 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? => ? = 1 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? => ? = 1 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? => ? = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? => ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? => ? = 2 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? => ? = 2 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? => ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? => ? = 2 + 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? => ? = 2 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? => ? = 1 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? => ? = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? => ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => ? => ? = 1 + 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? => ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => ? => ? = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => ? => ? = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => ? => ? = 1 + 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? => ? = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => ? => ? = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? => ? = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? => ? = 1 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? => ? = 1 + 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? => ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? => ? = 1 + 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => ? => ? = 1 + 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => ? => ? = 1 + 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => ? => ? = 1 + 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => ? => ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? => ? = 1 + 1
[[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => ? => ? = 1 + 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? => ? = 1 + 1
[[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? => ? = 0 + 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => ? => ? = 1 + 1
[[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? => ? = 1 + 1
[[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => ? => ? = 0 + 1
[[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => ? => ? = 0 + 1
[[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => ? => ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? => ? = 2 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? => ? = 2 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => ? => ? = 2 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? => ? = 1 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? => ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => ? => ? = 2 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => ? => ? = 1 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => ? => ? = 2 + 1
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.