Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
St000018: Permutations ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
[.,.]=>[1]=>0
[.,[.,.]]=>[2,1]=>1
[[.,.],.]=>[1,2]=>0
[.,[.,[.,.]]]=>[3,2,1]=>3
[.,[[.,.],.]]=>[2,3,1]=>2
[[.,.],[.,.]]=>[1,3,2]=>1
[[.,[.,.]],.]=>[2,1,3]=>1
[[[.,.],.],.]=>[1,2,3]=>0
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>6
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>5
[.,[[.,.],[.,.]]]=>[2,4,3,1]=>4
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>4
[.,[[[.,.],.],.]]=>[2,3,4,1]=>3
[[.,.],[.,[.,.]]]=>[1,4,3,2]=>3
[[.,.],[[.,.],.]]=>[1,3,4,2]=>2
[[.,[.,.]],[.,.]]=>[2,1,4,3]=>2
[[[.,.],.],[.,.]]=>[1,2,4,3]=>1
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>3
[[.,[[.,.],.]],.]=>[2,3,1,4]=>2
[[[.,.],[.,.]],.]=>[1,3,2,4]=>1
[[[.,[.,.]],.],.]=>[2,1,3,4]=>1
[[[[.,.],.],.],.]=>[1,2,3,4]=>0
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>10
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>9
[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>8
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>8
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>7
[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>7
[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>6
[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>6
[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>5
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>7
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>6
[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>5
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>5
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>4
[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>6
[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>5
[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>4
[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>4
[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>3
[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>4
[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>3
[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>3
[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>2
[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>4
[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>3
[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>2
[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>2
[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>1
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>6
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>5
[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>4
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>4
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>3
[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>3
[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>2
[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>2
[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>1
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>3
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>2
[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>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
[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>13
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>13
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>12
[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>12
[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>11
[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>11
[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>10
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>12
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>11
[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>10
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>10
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>9
[.,[[.,.],[.,[.,[.,.]]]]]=>[2,6,5,4,3,1]=>11
[.,[[.,.],[.,[[.,.],.]]]]=>[2,5,6,4,3,1]=>10
[.,[[.,.],[[.,.],[.,.]]]]=>[2,4,6,5,3,1]=>9
[.,[[.,.],[[.,[.,.]],.]]]=>[2,5,4,6,3,1]=>9
[.,[[.,.],[[[.,.],.],.]]]=>[2,4,5,6,3,1]=>8
[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>9
[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>8
[.,[[[.,.],.],[.,[.,.]]]]=>[2,3,6,5,4,1]=>8
[.,[[[.,.],.],[[.,.],.]]]=>[2,3,5,6,4,1]=>7
[.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>9
[.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>8
[.,[[[.,.],[.,.]],[.,.]]]=>[2,4,3,6,5,1]=>7
[.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>7
[.,[[[[.,.],.],.],[.,.]]]=>[2,3,4,6,5,1]=>6
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>11
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>10
[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>9
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>9
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>8
[.,[[[.,.],[.,[.,.]]],.]]=>[2,5,4,3,6,1]=>8
[.,[[[.,.],[[.,.],.]],.]]=>[2,4,5,3,6,1]=>7
[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>7
[.,[[[[.,.],.],[.,.]],.]]=>[2,3,5,4,6,1]=>6
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>8
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>7
[.,[[[[.,.],[.,.]],.],.]]=>[2,4,3,5,6,1]=>6
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>6
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>5
[[.,.],[.,[.,[.,[.,.]]]]]=>[1,6,5,4,3,2]=>10
[[.,.],[.,[.,[[.,.],.]]]]=>[1,5,6,4,3,2]=>9
[[.,.],[.,[[.,.],[.,.]]]]=>[1,4,6,5,3,2]=>8
[[.,.],[.,[[.,[.,.]],.]]]=>[1,5,4,6,3,2]=>8
[[.,.],[.,[[[.,.],.],.]]]=>[1,4,5,6,3,2]=>7
[[.,.],[[.,.],[.,[.,.]]]]=>[1,3,6,5,4,2]=>7
[[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2]=>6
[[.,.],[[.,[.,.]],[.,.]]]=>[1,4,3,6,5,2]=>6
[[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2]=>5
[[.,.],[[.,[.,[.,.]]],.]]=>[1,5,4,3,6,2]=>7
[[.,.],[[.,[[.,.],.]],.]]=>[1,4,5,3,6,2]=>6
[[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2]=>5
[[.,.],[[[.,[.,.]],.],.]]=>[1,4,3,5,6,2]=>5
[[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2]=>4
[[.,[.,.]],[.,[.,[.,.]]]]=>[2,1,6,5,4,3]=>7
[[.,[.,.]],[.,[[.,.],.]]]=>[2,1,5,6,4,3]=>6
[[.,[.,.]],[[.,.],[.,.]]]=>[2,1,4,6,5,3]=>5
[[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3]=>5
[[.,[.,.]],[[[.,.],.],.]]=>[2,1,4,5,6,3]=>4
[[[.,.],.],[.,[.,[.,.]]]]=>[1,2,6,5,4,3]=>6
[[[.,.],.],[.,[[.,.],.]]]=>[1,2,5,6,4,3]=>5
[[[.,.],.],[[.,.],[.,.]]]=>[1,2,4,6,5,3]=>4
[[[.,.],.],[[.,[.,.]],.]]=>[1,2,5,4,6,3]=>4
[[[.,.],.],[[[.,.],.],.]]=>[1,2,4,5,6,3]=>3
[[.,[.,[.,.]]],[.,[.,.]]]=>[3,2,1,6,5,4]=>6
[[.,[.,[.,.]]],[[.,.],.]]=>[3,2,1,5,6,4]=>5
[[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4]=>5
[[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4]=>4
[[[.,.],[.,.]],[.,[.,.]]]=>[1,3,2,6,5,4]=>4
[[[.,.],[.,.]],[[.,.],.]]=>[1,3,2,5,6,4]=>3
[[[.,[.,.]],.],[.,[.,.]]]=>[2,1,3,6,5,4]=>4
[[[.,[.,.]],.],[[.,.],.]]=>[2,1,3,5,6,4]=>3
[[[[.,.],.],.],[.,[.,.]]]=>[1,2,3,6,5,4]=>3
[[[[.,.],.],.],[[.,.],.]]=>[1,2,3,5,6,4]=>2
[[.,[.,[.,[.,.]]]],[.,.]]=>[4,3,2,1,6,5]=>7
[[.,[.,[[.,.],.]]],[.,.]]=>[3,4,2,1,6,5]=>6
[[.,[[.,.],[.,.]]],[.,.]]=>[2,4,3,1,6,5]=>5
[[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>5
[[.,[[[.,.],.],.]],[.,.]]=>[2,3,4,1,6,5]=>4
[[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5]=>4
[[[.,.],[[.,.],.]],[.,.]]=>[1,3,4,2,6,5]=>3
[[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>3
[[[[.,.],.],[.,.]],[.,.]]=>[1,2,4,3,6,5]=>2
[[[.,[.,[.,.]]],.],[.,.]]=>[3,2,1,4,6,5]=>4
[[[.,[[.,.],.]],.],[.,.]]=>[2,3,1,4,6,5]=>3
[[[[.,.],[.,.]],.],[.,.]]=>[1,3,2,4,6,5]=>2
[[[[.,[.,.]],.],.],[.,.]]=>[2,1,3,4,6,5]=>2
[[[[[.,.],.],.],.],[.,.]]=>[1,2,3,4,6,5]=>1
[[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>10
[[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>9
[[.,[.,[[.,.],[.,.]]]],.]=>[3,5,4,2,1,6]=>8
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>8
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>7
[[.,[[.,.],[.,[.,.]]]],.]=>[2,5,4,3,1,6]=>7
[[.,[[.,.],[[.,.],.]]],.]=>[2,4,5,3,1,6]=>6
[[.,[[.,[.,.]],[.,.]]],.]=>[3,2,5,4,1,6]=>6
[[.,[[[.,.],.],[.,.]]],.]=>[2,3,5,4,1,6]=>5
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>7
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>6
[[.,[[[.,.],[.,.]],.]],.]=>[2,4,3,5,1,6]=>5
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>5
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>4
[[[.,.],[.,[.,[.,.]]]],.]=>[1,5,4,3,2,6]=>6
[[[.,.],[.,[[.,.],.]]],.]=>[1,4,5,3,2,6]=>5
[[[.,.],[[.,.],[.,.]]],.]=>[1,3,5,4,2,6]=>4
[[[.,.],[[.,[.,.]],.]],.]=>[1,4,3,5,2,6]=>4
[[[.,.],[[[.,.],.],.]],.]=>[1,3,4,5,2,6]=>3
[[[.,[.,.]],[.,[.,.]]],.]=>[2,1,5,4,3,6]=>4
[[[.,[.,.]],[[.,.],.]],.]=>[2,1,4,5,3,6]=>3
[[[[.,.],.],[.,[.,.]]],.]=>[1,2,5,4,3,6]=>3
[[[[.,.],.],[[.,.],.]],.]=>[1,2,4,5,3,6]=>2
[[[.,[.,[.,.]]],[.,.]],.]=>[3,2,1,5,4,6]=>4
[[[.,[[.,.],.]],[.,.]],.]=>[2,3,1,5,4,6]=>3
[[[[.,.],[.,.]],[.,.]],.]=>[1,3,2,5,4,6]=>2
[[[[.,[.,.]],.],[.,.]],.]=>[2,1,3,5,4,6]=>2
[[[[[.,.],.],.],[.,.]],.]=>[1,2,3,5,4,6]=>1
[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>6
[[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>5
[[[.,[[.,.],[.,.]]],.],.]=>[2,4,3,1,5,6]=>4
[[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>4
[[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>3
[[[[.,.],[.,[.,.]]],.],.]=>[1,4,3,2,5,6]=>3
[[[[.,.],[[.,.],.]],.],.]=>[1,3,4,2,5,6]=>2
[[[[.,[.,.]],[.,.]],.],.]=>[2,1,4,3,5,6]=>2
[[[[[.,.],.],[.,.]],.],.]=>[1,2,4,3,5,6]=>1
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>3
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>2
[[[[[.,.],[.,.]],.],.],.]=>[1,3,2,4,5,6]=>1
[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>1
[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]=>[7,6,5,4,3,2,1]=>21
[.,[[.,[[[[.,.],.],.],.]],.]]=>[3,4,5,6,2,7,1]=>10
[.,[[[.,[[.,[.,.]],.]],.],.]]=>[4,3,5,2,6,7,1]=>10
[.,[[[.,[[[.,.],.],.]],.],.]]=>[3,4,5,2,6,7,1]=>9
[.,[[[[.,[.,[.,.]]],.],.],.]]=>[4,3,2,5,6,7,1]=>9
[.,[[[[.,[[.,.],.]],.],.],.]]=>[3,4,2,5,6,7,1]=>8
[.,[[[[[.,[.,.]],.],.],.],.]]=>[3,2,4,5,6,7,1]=>7
[.,[[[[[[.,.],.],.],.],.],.]]=>[2,3,4,5,6,7,1]=>6
[[.,.],[.,[.,[.,[.,[.,.]]]]]]=>[1,7,6,5,4,3,2]=>15
[[.,.],[.,[.,[.,[[.,.],.]]]]]=>[1,6,7,5,4,3,2]=>14
[[.,.],[.,[.,[[.,.],[.,.]]]]]=>[1,5,7,6,4,3,2]=>13
[[.,.],[.,[.,[[.,[.,.]],.]]]]=>[1,6,5,7,4,3,2]=>13
[[.,.],[.,[.,[[[.,.],.],.]]]]=>[1,5,6,7,4,3,2]=>12
[[.,.],[.,[[.,.],[.,[.,.]]]]]=>[1,4,7,6,5,3,2]=>12
[[.,.],[.,[[.,.],[[.,.],.]]]]=>[1,4,6,7,5,3,2]=>11
[[.,.],[.,[[.,[.,.]],[.,.]]]]=>[1,5,4,7,6,3,2]=>11
[[.,.],[.,[[.,[.,[.,.]]],.]]]=>[1,6,5,4,7,3,2]=>12
[[.,.],[.,[[.,[[.,.],.]],.]]]=>[1,5,6,4,7,3,2]=>11
[[.,.],[[.,.],[.,[.,[.,.]]]]]=>[1,3,7,6,5,4,2]=>11
[[.,.],[[.,[.,[.,[.,.]]]],.]]=>[1,6,5,4,3,7,2]=>11
[[.,[.,.]],[.,[.,[.,[.,.]]]]]=>[2,1,7,6,5,4,3]=>11
[[.,[.,[.,[.,[.,.]]]]],[.,.]]=>[5,4,3,2,1,7,6]=>11
[[.,[[[[.,.],.],.],.]],[.,.]]=>[2,3,4,5,1,7,6]=>5
[[[.,[[[.,.],.],.]],.],[.,.]]=>[2,3,4,1,5,7,6]=>4
[[[[[[.,.],.],.],.],.],[.,.]]=>[1,2,3,4,5,7,6]=>1
[[.,[.,[.,[.,[.,[.,.]]]]]],.]=>[6,5,4,3,2,1,7]=>15
[[.,[.,[.,[.,[[.,.],.]]]]],.]=>[5,6,4,3,2,1,7]=>14
[[.,[.,[.,[[.,.],[.,.]]]]],.]=>[4,6,5,3,2,1,7]=>13
[[.,[.,[.,[[.,[.,.]],.]]]],.]=>[5,4,6,3,2,1,7]=>13
[[.,[.,[.,[[[.,.],.],.]]]],.]=>[4,5,6,3,2,1,7]=>12
[[.,[.,[[.,.],[.,[.,.]]]]],.]=>[3,6,5,4,2,1,7]=>12
[[.,[.,[[.,.],[[.,.],.]]]],.]=>[3,5,6,4,2,1,7]=>11
[[.,[.,[[.,[.,.]],[.,.]]]],.]=>[4,3,6,5,2,1,7]=>11
[[.,[.,[[.,[.,[.,.]]],.]]],.]=>[5,4,3,6,2,1,7]=>12
[[.,[.,[[.,[[.,.],.]],.]]],.]=>[4,5,3,6,2,1,7]=>11
[[.,[[.,.],[.,[.,[.,.]]]]],.]=>[2,6,5,4,3,1,7]=>11
[[.,[[.,[.,[.,[.,.]]]],.]],.]=>[5,4,3,2,6,1,7]=>11
[[.,[[[[[.,.],.],.],.],.]],.]=>[2,3,4,5,6,1,7]=>5
[[[[[[.,.],.],.],.],[.,.]],.]=>[1,2,3,4,6,5,7]=>1
[[[.,[.,[.,[.,[.,.]]]]],.],.]=>[5,4,3,2,1,6,7]=>10
[[[.,[[[[.,.],.],.],.]],.],.]=>[2,3,4,5,1,6,7]=>4
[[[[.,[.,[.,[.,.]]]],.],.],.]=>[4,3,2,1,5,6,7]=>6
[[[[.,[[[.,.],.],.]],.],.],.]=>[2,3,4,1,5,6,7]=>3
[[[[[.,[.,[.,.]]],.],.],.],.]=>[3,2,1,4,5,6,7]=>3
[[[[[.,[[.,.],.]],.],.],.],.]=>[2,3,1,4,5,6,7]=>2
[[[[[[.,.],[.,.]],.],.],.],.]=>[1,3,2,4,5,6,7]=>1
[[[[[[.,[.,.]],.],.],.],.],.]=>[2,1,3,4,5,6,7]=>1
[[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7]=>0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[8,7,6,5,4,3,2,1]=>28
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]=>[7,6,8,5,4,3,2,1]=>26
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]=>[6,5,8,7,4,3,2,1]=>24
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]=>[7,6,5,4,8,3,2,1]=>24
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]=>[6,5,7,4,8,3,2,1]=>22
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]=>[3,5,8,7,6,4,2,1]=>20
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]=>[4,3,6,5,8,7,2,1]=>16
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]=>[2,6,8,7,5,4,3,1]=>20
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]=>[2,3,5,6,8,7,4,1]=>12
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]=>[3,2,6,5,4,8,7,1]=>12
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]=>[7,6,5,4,3,2,8,1]=>22
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]=>[6,5,7,4,3,2,8,1]=>20
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]=>[6,5,4,3,7,2,8,1]=>18
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]=>[5,4,6,3,7,2,8,1]=>16
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]=>[4,3,6,5,7,2,8,1]=>14
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]=>[3,2,6,5,7,4,8,1]=>12
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]=>[4,3,5,2,7,6,8,1]=>12
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]=>[3,2,5,4,7,6,8,1]=>10
[.,[[[[.,[[[.,.],.],.]],.],.],.]]=>[3,4,5,2,6,7,8,1]=>10
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]=>[4,3,2,5,6,7,8,1]=>10
[.,[[[[[.,[[.,.],.]],.],.],.],.]]=>[3,4,2,5,6,7,8,1]=>9
[.,[[[[[[.,[.,.]],.],.],.],.],.]]=>[3,2,4,5,6,7,8,1]=>8
[.,[[[[[[[.,.],.],.],.],.],.],.]]=>[2,3,4,5,6,7,8,1]=>7
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[1,8,7,6,5,4,3,2]=>21
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]=>[1,7,8,6,5,4,3,2]=>20
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]=>[1,6,8,7,5,4,3,2]=>19
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]=>[1,7,6,8,5,4,3,2]=>19
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]=>[1,6,7,8,5,4,3,2]=>18
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]=>[1,5,8,7,6,4,3,2]=>18
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]=>[1,7,6,5,8,4,3,2]=>18
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]=>[1,3,5,7,8,6,4,2]=>12
[[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]=>[1,4,3,8,7,6,5,2]=>13
[[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]=>[1,6,5,4,3,8,7,2]=>13
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]=>[1,4,3,6,5,8,7,2]=>9
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>[2,1,8,7,6,5,4,3]=>16
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]=>[2,1,6,5,7,4,8,3]=>10
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]=>[2,1,5,4,7,6,8,3]=>8
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]=>[1,2,6,7,8,5,4,3]=>12
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]=>[3,2,1,8,7,6,5,4]=>13
[[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]=>[3,2,1,6,5,8,7,4]=>9
[[[[.,.],.],.],[[[[.,.],.],.],.]]=>[1,2,3,5,6,7,8,4]=>4
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]=>[4,3,2,1,8,7,6,5]=>12
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]=>[3,4,2,1,7,8,6,5]=>10
[[.,[[.,.],[.,.]]],[[.,.],[.,.]]]=>[2,4,3,1,6,8,7,5]=>8
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]=>[2,4,3,1,7,6,8,5]=>8
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]=>[3,2,4,1,6,8,7,5]=>8
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]=>[3,2,4,1,7,6,8,5]=>8
[[.,[[[.,.],.],.]],[[[.,.],.],.]]=>[2,3,4,1,6,7,8,5]=>6
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]=>[2,1,4,3,8,7,6,5]=>8
[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]=>[2,1,4,3,6,8,7,5]=>6
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]=>[2,1,4,3,7,6,8,5]=>6
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]=>[5,4,3,2,1,8,7,6]=>13
[[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]=>[3,2,5,4,1,8,7,6]=>9
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]=>[3,2,1,4,5,8,7,6]=>6
[[[[.,[[.,.],.]],.],.],[[.,.],.]]=>[2,3,1,4,5,7,8,6]=>4
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]=>[6,5,4,3,2,1,8,7]=>16
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]=>[4,3,5,2,6,1,8,7]=>10
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]=>[3,2,5,4,6,1,8,7]=>8
[[.,[[[[[.,.],.],.],.],.]],[.,.]]=>[2,3,4,5,6,1,8,7]=>6
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]=>[2,1,6,5,4,3,8,7]=>8
[[[.,[.,.]],[[.,.],[.,.]]],[.,.]]=>[2,1,4,6,5,3,8,7]=>6
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]=>[2,1,5,4,6,3,8,7]=>6
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]=>[4,3,2,1,6,5,8,7]=>8
[[[.,[[.,.],[.,.]]],[.,.]],[.,.]]=>[2,4,3,1,6,5,8,7]=>6
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]=>[3,2,4,1,6,5,8,7]=>6
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5,8,7]=>4
[[[[[[.,[.,.]],.],.],.],.],[.,.]]=>[2,1,3,4,5,6,8,7]=>2
[[[[[[[.,.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,8,7]=>1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]=>[7,6,5,4,3,2,1,8]=>21
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]=>[6,7,5,4,3,2,1,8]=>20
[[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]=>[5,7,6,4,3,2,1,8]=>19
[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]=>[6,5,7,4,3,2,1,8]=>19
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]=>[5,6,7,4,3,2,1,8]=>18
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]=>[4,7,6,5,3,2,1,8]=>18
[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]=>[6,5,4,7,3,2,1,8]=>18
[[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]=>[3,2,7,6,5,4,1,8]=>13
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]=>[5,4,3,2,7,6,1,8]=>13
[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]=>[3,2,5,4,7,6,1,8]=>9
[[.,[[[[[[.,.],.],.],.],.],.]],.]=>[2,3,4,5,6,7,1,8]=>6
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]=>[1,4,3,2,7,6,5,8]=>6
[[[[.,.],[[.,.],.]],[[.,.],.]],.]=>[1,3,4,2,6,7,5,8]=>4
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]=>[1,3,2,5,4,7,6,8]=>3
[[[[[[.,.],.],.],[.,.]],[.,.]],.]=>[1,2,3,5,4,7,6,8]=>2
[[[[[[.,.],[.,.]],.],.],[.,.]],.]=>[1,3,2,4,5,7,6,8]=>2
[[[[[[[.,.],.],.],.],.],[.,.]],.]=>[1,2,3,4,5,7,6,8]=>1
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]=>[6,5,4,3,2,1,7,8]=>15
[[[.,[[[[[.,.],.],.],.],.]],.],.]=>[2,3,4,5,6,1,7,8]=>5
[[[[[[.,.],.],[.,.]],[.,.]],.],.]=>[1,2,4,3,6,5,7,8]=>2
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]=>[5,4,3,2,1,6,7,8]=>10
[[[[.,[[[[.,.],.],.],.]],.],.],.]=>[2,3,4,5,1,6,7,8]=>4
[[[[[[.,.],[.,.]],[.,.]],.],.],.]=>[1,3,2,5,4,6,7,8]=>2
[[[[[[[.,.],.],.],[.,.]],.],.],.]=>[1,2,3,5,4,6,7,8]=>1
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]=>[4,3,2,1,5,6,7,8]=>6
[[[[[[.,[.,[.,.]]],.],.],.],.],.]=>[3,2,1,4,5,6,7,8]=>3
[[[[[[[.,.],[.,.]],.],.],.],.],.]=>[1,3,2,4,5,6,7,8]=>1
[[[[[[[.,[.,.]],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8]=>1
[[[[[[[[.,.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8]=>0
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[9,8,7,6,5,4,3,2,1]=>36
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]=>[2,3,4,5,6,7,8,9,1]=>8
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8,9]=>0
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[10,9,8,7,6,5,4,3,2,1]=>45
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]=>[2,3,4,5,6,7,8,9,10,1]=>9
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8,9,10]=>0
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]=>[8,7,6,5,4,3,2,1,9]=>28
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]=>[9,8,7,6,5,4,3,2,1,10]=>36
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5,8,7,10,9]=>5
[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>6
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,9,8]=>1
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,8,10,9]=>1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]=>[3,2,4,5,6,7,8,9,1]=>9
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,8,9,11,10]=>1
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]=>[3,4,2,5,6,7,8,9,1]=>10
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]=>[3,2,4,5,6,7,8,9,10,1]=>10
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]=>[2,3,4,5,6,7,8,9,10,11,1]=>10
[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[1,9,8,7,6,5,4,3,2]=>28
[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]=>[7,8,6,5,4,3,2,1,9]=>27
[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]=>[1,8,9,7,6,5,4,3,2]=>27
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[1,10,9,8,7,6,5,4,3,2]=>36
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]=>[10,9,8,7,6,5,4,3,2,1,11]=>45
[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]=>[8,9,7,6,5,4,3,2,1,10]=>35
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]=>[7,6,8,5,4,3,2,1,9]=>26
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]=>[1,8,7,9,6,5,4,3,2]=>26
[[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]=>[1,9,10,8,7,6,5,4,3,2]=>35
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[1,11,10,9,8,7,6,5,4,3,2]=>45
[[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[2,1,10,9,8,7,6,5,4,3]=>29
[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]=>[4,3,2,1,10,9,8,7,6,5]=>21
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]=>[6,5,4,3,2,1,10,9,8,7]=>21
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]=>[8,7,6,5,4,3,2,1,10,9]=>29
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]=>[7,6,5,4,3,2,1,8,9]=>21
[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]=>[6,5,4,3,2,1,7,8,9]=>15
[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]=>[5,4,3,2,1,6,7,8,9]=>10
[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]=>[4,3,2,1,5,6,7,8,9]=>6
[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]=>[3,2,1,4,5,6,7,8,9]=>3
[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9]=>1
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]=>[8,7,6,5,4,3,2,1,9,10]=>28
[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]=>[7,6,5,4,3,2,1,8,9,10]=>21
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]=>[6,5,4,3,2,1,7,8,9,10]=>15
[[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]=>[5,4,3,2,1,6,7,8,9,10]=>10
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]=>[4,3,2,1,5,6,7,8,9,10]=>6
[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]=>[3,2,1,4,5,6,7,8,9,10]=>3
[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9,10]=>1
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9,10,11]=>1
[[.,[[[[[[[.,.],.],.],.],.],.],.]],.]=>[2,3,4,5,6,7,8,1,9]=>7
[[[.,[[[[[[.,.],.],.],.],.],.]],.],.]=>[2,3,4,5,6,7,1,8,9]=>6
[[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]=>[2,3,4,5,6,7,8,9,1,10]=>8
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Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
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.
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.
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