Identifier
Values
[.,.] => [1] => 1
[.,[.,.]] => [2,1] => 1
[[.,.],.] => [1,2] => 1
[.,[.,[.,.]]] => [3,2,1] => 1
[.,[[.,.],.]] => [2,3,1] => 1
[[.,.],[.,.]] => [1,3,2] => 2
[[.,[.,.]],.] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => 1
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => 2
[.,[[.,[.,.]],.]] => [3,2,4,1] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => 5
[[.,.],[[.,.],.]] => [1,3,4,2] => 3
[[.,[.,.]],[.,.]] => [2,1,4,3] => 3
[[[.,.],.],[.,.]] => [1,2,4,3] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => 1
[[[.,.],[.,.]],.] => [1,3,2,4] => 2
[[[.,[.,.]],.],.] => [2,1,3,4] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 5
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 3
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 3
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 3
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 2
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 14
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 9
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 11
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 7
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 4
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 14
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 6
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 14
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 6
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 4
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 4
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 8
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 4
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 1
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 2
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 5
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 3
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 3
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 3
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 1
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 1
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 2
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 5
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 3
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 3
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 2
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 14
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 9
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 11
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 7
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 4
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 14
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 6
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 14
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 6
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 4
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 4
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 8
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 4
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 4
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 2
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 5
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => 3
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 3
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 3
>>> Load all 242 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 1
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 1
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => 2
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 1
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 42
[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 28
[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 37
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => 23
[[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => 14
[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 58
[[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 26
[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => 34
[[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 19
[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => 19
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => 12
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => 14
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => 9
[[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => 5
[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => 84
[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => 40
[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 35
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => 25
[[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => 10
[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 84
[[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 40
[[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 35
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => 25
[[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => 10
[[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 30
[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => 10
[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => 30
[[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => 10
[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => 60
[[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => 20
[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => 30
[[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => 10
[[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 30
[[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => 10
[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => 5
[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => 5
[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => 10
[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => 5
[[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => 5
[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => 25
[[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => 15
[[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => 15
[[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => 15
[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => 5
[[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => 5
[[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => 10
[[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => 5
[[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 1
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 1
[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => 2
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 1
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 1
[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => 5
[[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => 3
[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => 3
[[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => 3
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 1
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 1
[[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => 2
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 1
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 1
[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => 14
[[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => 9
[[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => 11
[[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => 7
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => 4
[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => 14
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => 6
[[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => 14
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => 6
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => 4
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => 4
[[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => 8
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => 4
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => 4
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 1
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 1
[[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => 2
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 1
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 1
[[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => 5
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => 3
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => 3
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => 3
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 1
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 1
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => 2
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 1
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 1
[.,[[.,[[[[.,.],.],.],.]],.]] => [3,4,5,6,2,7,1] => 1
[.,[[[.,[[.,[.,.]],.]],.],.]] => [4,3,5,2,6,7,1] => 1
[.,[[[.,[[[.,.],.],.]],.],.]] => [3,4,5,2,6,7,1] => 1
[.,[[[[.,[.,[.,.]]],.],.],.]] => [4,3,2,5,6,7,1] => 1
[.,[[[[.,[[.,.],.]],.],.],.]] => [3,4,2,5,6,7,1] => 1
[.,[[[[[.,[.,.]],.],.],.],.]] => [3,2,4,5,6,7,1] => 1
[.,[[[[[[.,.],.],.],.],.],.]] => [2,3,4,5,6,7,1] => 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]] => [1,7,6,5,4,3,2] => 132
[[.,.],[.,[.,[.,[[.,.],.]]]]] => [1,6,7,5,4,3,2] => 90
[[.,.],[.,[.,[[.,.],[.,.]]]]] => [1,5,7,6,4,3,2] => 124
[[.,.],[.,[.,[[.,[.,.]],.]]]] => [1,6,5,7,4,3,2] => 76
[[.,.],[.,[.,[[[.,.],.],.]]]] => [1,5,6,7,4,3,2] => 48
[[.,.],[.,[[.,.],[.,[.,.]]]]] => [1,4,7,6,5,3,2] => 211
[[.,.],[.,[[.,.],[[.,.],.]]]] => [1,4,6,7,5,3,2] => 97
[[.,.],[.,[[.,[.,.]],[.,.]]]] => [1,5,4,7,6,3,2] => 125
[[.,.],[.,[[.,[.,[.,.]]],.]]] => [1,6,5,4,7,3,2] => 66
[[.,.],[.,[[.,[[.,.],.]],.]]] => [1,5,6,4,7,3,2] => 43
[[.,.],[[.,.],[.,[.,[.,.]]]]] => [1,3,7,6,5,4,2] => 378
[[.,.],[[.,[.,[.,[.,.]]]],.]] => [1,6,5,4,3,7,2] => 56
[[.,[.,.]],[.,[.,[.,[.,.]]]]] => [2,1,7,6,5,4,3] => 594
[[.,[.,[.,[.,[.,.]]]]],[.,.]] => [5,4,3,2,1,7,6] => 6
[[.,[[[[.,.],.],.],.]],[.,.]] => [2,3,4,5,1,7,6] => 6
[[[.,[[[.,.],.],.]],.],[.,.]] => [2,3,4,1,5,7,6] => 6
[[[[[[.,.],.],.],.],.],[.,.]] => [1,2,3,4,5,7,6] => 6
[[.,[.,[.,[.,[[.,.],.]]]]],.] => [5,6,4,3,2,1,7] => 1
[[.,[.,[.,[[.,.],[.,.]]]]],.] => [4,6,5,3,2,1,7] => 2
[[.,[.,[.,[[.,[.,.]],.]]]],.] => [5,4,6,3,2,1,7] => 1
[[.,[.,[.,[[[.,.],.],.]]]],.] => [4,5,6,3,2,1,7] => 1
[[.,[.,[[.,.],[.,[.,.]]]]],.] => [3,6,5,4,2,1,7] => 5
[[.,[.,[[.,.],[[.,.],.]]]],.] => [3,5,6,4,2,1,7] => 3
[[.,[.,[[.,[.,.]],[.,.]]]],.] => [4,3,6,5,2,1,7] => 3
[[.,[.,[[.,[.,[.,.]]],.]]],.] => [5,4,3,6,2,1,7] => 1
[[.,[.,[[.,[[.,.],.]],.]]],.] => [4,5,3,6,2,1,7] => 1
[[.,[[.,.],[.,[.,[.,.]]]]],.] => [2,6,5,4,3,1,7] => 14
[[.,[[.,[.,[.,[.,.]]]],.]],.] => [5,4,3,2,6,1,7] => 1
[[.,[[[[[.,.],.],.],.],.]],.] => [2,3,4,5,6,1,7] => 1
[[[[[[.,.],.],.],.],[.,.]],.] => [1,2,3,4,6,5,7] => 5
[[[.,[.,[.,[.,[.,.]]]]],.],.] => [5,4,3,2,1,6,7] => 1
[[[.,[[[[.,.],.],.],.]],.],.] => [2,3,4,5,1,6,7] => 1
[[[[.,[.,[.,[.,.]]]],.],.],.] => [4,3,2,1,5,6,7] => 1
[[[[.,[[[.,.],.],.]],.],.],.] => [2,3,4,1,5,6,7] => 1
[[[[[.,[.,[.,.]]],.],.],.],.] => [3,2,1,4,5,6,7] => 1
[[[[[.,[[.,.],.]],.],.],.],.] => [2,3,1,4,5,6,7] => 1
[[[[[[.,.],[.,.]],.],.],.],.] => [1,3,2,4,5,6,7] => 2
[[[[[[.,[.,.]],.],.],.],.],.] => [2,1,3,4,5,6,7] => 1
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => 1
search for individual values
searching the database for the individual values of this statistic
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Description
The number of reduced Kogan faces with the permutation as type.
This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
Map
to 312-avoiding permutation
Description
Return a 312-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 minimal element of this Sylvester class.