Identifier
Values
[.,.] => [1] => 0
[.,[.,.]] => [2,1] => 1
[[.,.],.] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => 3
[.,[[.,.],.]] => [2,3,1] => 2
[[.,.],[.,.]] => [3,1,2] => 1
[[.,[.,.]],.] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 6
[.,[.,[[.,.],.]]] => [3,4,2,1] => 5
[.,[[.,.],[.,.]]] => [4,2,3,1] => 4
[.,[[.,[.,.]],.]] => [3,2,4,1] => 4
[.,[[[.,.],.],.]] => [2,3,4,1] => 3
[[.,.],[.,[.,.]]] => [4,3,1,2] => 3
[[.,.],[[.,.],.]] => [3,4,1,2] => 2
[[.,[.,.]],[.,.]] => [4,2,1,3] => 3
[[[.,.],.],[.,.]] => [4,1,2,3] => 1
[[.,[.,[.,.]]],.] => [3,2,1,4] => 3
[[.,[[.,.],.]],.] => [2,3,1,4] => 2
[[[.,.],[.,.]],.] => [3,1,2,4] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 10
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 9
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 8
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 8
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 7
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 7
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 6
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 7
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 5
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 7
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 6
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 5
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 5
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 4
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 6
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 5
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 4
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 4
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 3
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 6
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 5
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 3
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 6
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 4
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 3
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 3
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 6
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 5
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 4
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 4
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 3
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 3
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 2
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 3
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 1
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 3
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 2
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 15
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 14
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 13
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 13
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 12
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 12
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 11
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 12
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 10
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 12
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 11
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 10
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 10
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 9
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 11
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 10
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 9
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 9
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 8
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 11
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 10
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 8
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 7
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 11
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 9
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 8
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 8
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 6
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 11
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 10
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 9
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 9
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 8
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 8
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 7
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 8
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 6
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 8
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 7
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => 6
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 6
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 5
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => 10
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => 9
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => 8
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => 8
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => 7
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 7
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 6
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 7
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 5
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 7
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => 6
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => 5
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => 5
[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => 4
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => 10
[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => 9
[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => 8
[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => 8
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => 7
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 6
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 5
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 4
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 4
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 3
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => 10
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => 9
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => 7
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => 6
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 6
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 5
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => 6
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => 5
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 3
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 2
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => 10
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => 8
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => 7
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => 7
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => 5
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 6
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 4
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => 6
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 3
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => 6
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => 4
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 3
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => 3
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 10
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 9
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => 8
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 8
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 7
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => 7
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => 6
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => 7
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => 5
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 7
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 6
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => 5
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 5
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 4
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 6
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 5
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 4
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 4
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 3
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => 6
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => 5
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 3
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 2
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => 6
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => 4
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 3
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => 3
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 1
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 6
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 5
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => 4
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 4
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 3
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 3
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 2
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => 3
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 1
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 3
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 2
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => 1
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 1
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 0
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Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called Mahonian statistic.
Map
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.