Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000568
(load all 3 compositions to match this statistic)
Mapping
St000568: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => 1
[[.,.],.]
 => 1
[.,[.,[.,.]]]
 => 1
[.,[[.,.],.]]
 => 2
[[.,.],[.,.]]
 => 1
[[.,[.,.]],.]
 => 2
[[[.,.],.],.]
 => 1
[.,[.,[.,[.,.]]]]
 => 1
[.,[.,[[.,.],.]]]
 => 2
[.,[[.,.],[.,.]]]
 => 2
[.,[[.,[.,.]],.]]
 => 2
[.,[[[.,.],.],.]]
 => 2
[[.,.],[.,[.,.]]]
 => 1
[[.,.],[[.,.],.]]
 => 2
[[.,[.,.]],[.,.]]
 => 2
[[[.,.],.],[.,.]]
 => 1
[[.,[.,[.,.]]],.]
 => 2
[[.,[[.,.],.]],.]
 => 2
[[[.,.],[.,.]],.]
 => 2
[[[.,[.,.]],.],.]
 => 2
[[[[.,.],.],.],.]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => 2
[.,[.,[[.,.],[.,.]]]]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => 2
[.,[.,[[[.,.],.],.]]]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => 2
[.,[[.,.],[[.,.],.]]]
 => 3
[.,[[.,[.,.]],[.,.]]]
 => 2
[.,[[[.,.],.],[.,.]]]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => 2
[.,[[.,[[.,.],.]],.]]
 => 3
[.,[[[.,.],[.,.]],.]]
 => 2
[.,[[[.,[.,.]],.],.]]
 => 3
[.,[[[[.,.],.],.],.]]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => 1
[[.,.],[.,[[.,.],.]]]
 => 2
[[.,.],[[.,.],[.,.]]]
 => 2
[[.,.],[[.,[.,.]],.]]
 => 2
[[.,.],[[[.,.],.],.]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => 2
[[.,[.,.]],[[.,.],.]]
 => 3
[[[.,.],.],[.,[.,.]]]
 => 1
[[[.,.],.],[[.,.],.]]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => 2
[[[[.,.],.],.],[.,.]]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => 2
Description
The hook number of a binary tree. A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
Matching statistic: St000758
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
Mp00178: Binary words —to composition⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[.,[.,.]]
 => [1,0,1,0]
 => 1010 => [1,2,2] => 2 = 1 + 1
[[.,.],.]
 => [1,1,0,0]
 => 1100 => [1,1,3] => 2 = 1 + 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 101010 => [1,2,2,2] => 2 = 1 + 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 101100 => [1,2,1,3] => 3 = 2 + 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 110010 => [1,1,3,2] => 2 = 1 + 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 110100 => [1,1,2,3] => 3 = 2 + 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 111000 => [1,1,1,4] => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => [1,2,2,2,2] => 2 = 1 + 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => [1,2,2,1,3] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => [1,2,1,3,2] => 3 = 2 + 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => [1,2,1,2,3] => 3 = 2 + 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => [1,2,1,1,4] => 3 = 2 + 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => [1,1,3,2,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => [1,1,3,1,3] => 3 = 2 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => [1,1,2,3,2] => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => [1,1,1,4,2] => 2 = 1 + 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => [1,1,2,2,3] => 3 = 2 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => [1,1,2,1,4] => 3 = 2 + 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => [1,1,1,3,3] => 3 = 2 + 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => [1,1,1,2,4] => 3 = 2 + 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => [1,1,1,1,5] => 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => [1,2,2,2,2,2] => ? = 1 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => [1,2,2,2,1,3] => ? = 2 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => [1,2,2,1,3,2] => ? = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => [1,2,2,1,2,3] => ? = 2 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => [1,2,2,1,1,4] => ? = 2 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => [1,2,1,3,2,2] => ? = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => [1,2,1,3,1,3] => ? = 3 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => [1,2,1,2,3,2] => ? = 2 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => [1,2,1,1,4,2] => ? = 2 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => [1,2,1,2,2,3] => ? = 2 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => [1,2,1,2,1,4] => ? = 3 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => [1,2,1,1,3,3] => ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => [1,2,1,1,2,4] => ? = 3 + 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => [1,2,1,1,1,5] => ? = 2 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => [1,1,3,2,2,2] => ? = 1 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => [1,1,3,2,1,3] => ? = 2 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => [1,1,3,1,3,2] => ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => [1,1,3,1,2,3] => ? = 2 + 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => [1,1,3,1,1,4] => ? = 2 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => [1,1,2,3,2,2] => ? = 2 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => [1,1,2,3,1,3] => ? = 3 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => [1,1,1,4,2,2] => ? = 1 + 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => [1,1,1,4,1,3] => ? = 2 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => [1,1,2,2,3,2] => ? = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => [1,1,2,1,4,2] => ? = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => [1,1,1,3,3,2] => ? = 2 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => [1,1,1,2,4,2] => ? = 2 + 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => [1,1,1,1,5,2] => ? = 1 + 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => [1,1,2,2,2,3] => ? = 2 + 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => [1,1,2,2,1,4] => ? = 3 + 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => [1,1,2,1,3,3] => ? = 2 + 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => [1,1,2,1,2,4] => ? = 3 + 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => [1,1,2,1,1,5] => ? = 2 + 1
[[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => [1,1,1,3,2,3] => ? = 2 + 1
[[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => [1,1,1,3,1,4] => ? = 2 + 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => [1,1,1,2,3,3] => ? = 3 + 1
[[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => [1,1,1,1,4,3] => ? = 2 + 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => [1,1,1,2,2,4] => ? = 2 + 1
[[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => [1,1,1,2,1,5] => ? = 2 + 1
[[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => [1,1,1,1,3,4] => ? = 2 + 1
[[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => [1,1,1,1,2,5] => ? = 2 + 1
[[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 1 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => [1,2,2,2,2,2,2] => ? = 1 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => [1,2,2,2,2,1,3] => ? = 2 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => [1,2,2,2,1,3,2] => ? = 2 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => [1,2,2,2,1,2,3] => ? = 2 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => [1,2,2,2,1,1,4] => ? = 2 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => [1,2,2,1,3,2,2] => ? = 2 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => [1,2,2,1,3,1,3] => ? = 3 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => [1,2,2,1,2,3,2] => ? = 2 + 1
Description
The length of the longest staircase fitting into an integer composition. For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St001960
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[.,[.,.]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [3,1,2] => 0 = 1 - 1
[[.,.],.]
 => [1,1,0,0]
 => [1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1 = 2 - 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1 = 2 - 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1 = 2 - 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1 = 2 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1 = 2 - 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 2 - 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 2 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ? = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 3 - 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ? = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 2 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 3 - 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 3 - 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ? = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ? = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ? = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ? = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ? = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ? = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ? = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ? = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ? = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ? = 2 - 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 1 - 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 2 - 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 3 - 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 2 - 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 3 - 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 - 1
[[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
[[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 2 - 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 3 - 1
[[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 2 - 1
[[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 - 1
[[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 2 - 1
[[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 2 - 1
[[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 2 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 2 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 2 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 2 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 2 - 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 3 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 2 - 1
Description
The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].