Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
St001558: 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
[[.,.],[.,[[[.,.],.],[.,.]]]]=>[1,4,5,7,6,3,2]=>10
[[.,.],[.,[[[[.,.],.],.],.]]]=>[1,4,5,6,7,3,2]=>9
[[.,.],[[.,.],[.,[.,[.,.]]]]]=>[1,3,7,6,5,4,2]=>11
[[.,.],[[.,.],[.,[[.,.],.]]]]=>[1,3,6,7,5,4,2]=>10
[[.,.],[[.,.],[[.,.],[.,.]]]]=>[1,3,5,7,6,4,2]=>9
[[.,.],[[.,.],[[.,[.,.]],.]]]=>[1,3,6,5,7,4,2]=>9
[[.,.],[[.,.],[[[.,.],.],.]]]=>[1,3,5,6,7,4,2]=>8
[[.,.],[[.,[.,.]],[.,[.,.]]]]=>[1,4,3,7,6,5,2]=>9
[[.,.],[[.,[.,.]],[[.,.],.]]]=>[1,4,3,6,7,5,2]=>8
[[.,.],[[[.,.],.],[.,[.,.]]]]=>[1,3,4,7,6,5,2]=>8
[[.,.],[[[.,.],.],[[.,.],.]]]=>[1,3,4,6,7,5,2]=>7
[[.,.],[[.,[[.,.],.]],[.,.]]]=>[1,4,5,3,7,6,2]=>8
[[.,.],[[[.,.],[.,.]],[.,.]]]=>[1,3,5,4,7,6,2]=>7
[[.,.],[[[.,[.,.]],.],[.,.]]]=>[1,4,3,5,7,6,2]=>7
[[.,.],[[[[.,.],.],.],[.,.]]]=>[1,3,4,5,7,6,2]=>6
[[.,.],[[.,[[[.,.],.],.]],.]]=>[1,4,5,6,3,7,2]=>8
[[.,.],[[[.,.],[.,[.,.]]],.]]=>[1,3,6,5,4,7,2]=>8
[[.,.],[[[.,.],[[.,.],.]],.]]=>[1,3,5,6,4,7,2]=>7
[[.,.],[[[.,[.,.]],[.,.]],.]]=>[1,4,3,6,5,7,2]=>7
[[.,.],[[[[.,.],.],[.,.]],.]]=>[1,3,4,6,5,7,2]=>6
[[.,.],[[[.,[[.,.],.]],.],.]]=>[1,4,5,3,6,7,2]=>7
[[.,.],[[[[.,.],[.,.]],.],.]]=>[1,3,5,4,6,7,2]=>6
[[.,.],[[[[.,[.,.]],.],.],.]]=>[1,4,3,5,6,7,2]=>6
[[.,.],[[[[[.,.],.],.],.],.]]=>[1,3,4,5,6,7,2]=>5
[[[.,.],.],[.,[.,[.,[.,.]]]]]=>[1,2,7,6,5,4,3]=>10
[[[.,.],.],[.,[.,[[.,.],.]]]]=>[1,2,6,7,5,4,3]=>9
[[[.,.],.],[.,[[.,.],[.,.]]]]=>[1,2,5,7,6,4,3]=>8
[[[.,.],.],[.,[[.,[.,.]],.]]]=>[1,2,6,5,7,4,3]=>8
[[[.,.],.],[.,[[[.,.],.],.]]]=>[1,2,5,6,7,4,3]=>7
[[[.,.],.],[[.,.],[.,[.,.]]]]=>[1,2,4,7,6,5,3]=>7
[[[.,.],.],[[.,.],[[.,.],.]]]=>[1,2,4,6,7,5,3]=>6
[[[.,.],.],[[.,[.,.]],[.,.]]]=>[1,2,5,4,7,6,3]=>6
[[[.,.],.],[[[.,.],.],[.,.]]]=>[1,2,4,5,7,6,3]=>5
[[[.,.],.],[[.,[.,[.,.]]],.]]=>[1,2,6,5,4,7,3]=>7
[[[.,.],.],[[.,[[.,.],.]],.]]=>[1,2,5,6,4,7,3]=>6
[[[.,.],.],[[[.,.],[.,.]],.]]=>[1,2,4,6,5,7,3]=>5
[[[.,.],.],[[[.,[.,.]],.],.]]=>[1,2,5,4,6,7,3]=>5
[[[.,.],.],[[[[.,.],.],.],.]]=>[1,2,4,5,6,7,3]=>4
[[[.,.],[.,.]],[.,[.,[.,.]]]]=>[1,3,2,7,6,5,4]=>7
[[[.,.],[.,.]],[.,[[.,.],.]]]=>[1,3,2,6,7,5,4]=>6
[[[.,.],[.,.]],[[.,.],[.,.]]]=>[1,3,2,5,7,6,4]=>5
[[[.,.],[.,.]],[[.,[.,.]],.]]=>[1,3,2,6,5,7,4]=>5
[[[.,.],[.,.]],[[[.,.],.],.]]=>[1,3,2,5,6,7,4]=>4
[[[[.,.],.],.],[.,[.,[.,.]]]]=>[1,2,3,7,6,5,4]=>6
[[[[.,.],.],.],[.,[[.,.],.]]]=>[1,2,3,6,7,5,4]=>5
[[[[.,.],.],.],[[.,.],[.,.]]]=>[1,2,3,5,7,6,4]=>4
[[[[.,.],.],.],[[.,[.,.]],.]]=>[1,2,3,6,5,7,4]=>4
[[[[.,.],.],.],[[[.,.],.],.]]=>[1,2,3,5,6,7,4]=>3
[[[.,.],[.,[.,.]]],[.,[.,.]]]=>[1,4,3,2,7,6,5]=>6
[[[.,.],[.,[.,.]]],[[.,.],.]]=>[1,4,3,2,6,7,5]=>5
[[[.,.],[[.,.],.]],[.,[.,.]]]=>[1,3,4,2,7,6,5]=>5
[[[.,.],[[.,.],.]],[[.,.],.]]=>[1,3,4,2,6,7,5]=>4
[[[[.,.],.],[.,.]],[.,[.,.]]]=>[1,2,4,3,7,6,5]=>4
[[[[.,.],.],[.,.]],[[.,.],.]]=>[1,2,4,3,6,7,5]=>3
[[[[.,.],[.,.]],.],[.,[.,.]]]=>[1,3,2,4,7,6,5]=>4
[[[[.,.],[.,.]],.],[[.,.],.]]=>[1,3,2,4,6,7,5]=>3
[[[[[.,.],.],.],.],[.,[.,.]]]=>[1,2,3,4,7,6,5]=>3
[[[[[.,.],.],.],.],[[.,.],.]]=>[1,2,3,4,6,7,5]=>2
[[[.,.],[.,[[.,.],.]]],[.,.]]=>[1,4,5,3,2,7,6]=>6
[[[.,.],[[.,.],[.,.]]],[.,.]]=>[1,3,5,4,2,7,6]=>5
[[[.,.],[[.,[.,.]],.]],[.,.]]=>[1,4,3,5,2,7,6]=>5
[[[.,.],[[[.,.],.],.]],[.,.]]=>[1,3,4,5,2,7,6]=>4
[[[[.,.],.],[.,[.,.]]],[.,.]]=>[1,2,5,4,3,7,6]=>4
[[[[.,.],.],[[.,.],.]],[.,.]]=>[1,2,4,5,3,7,6]=>3
[[[[.,.],[.,.]],[.,.]],[.,.]]=>[1,3,2,5,4,7,6]=>3
[[[[[.,.],.],.],[.,.]],[.,.]]=>[1,2,3,5,4,7,6]=>2
[[[[.,.],[.,[.,.]]],.],[.,.]]=>[1,4,3,2,5,7,6]=>4
[[[[.,.],[[.,.],.]],.],[.,.]]=>[1,3,4,2,5,7,6]=>3
[[[[[.,.],.],[.,.]],.],[.,.]]=>[1,2,4,3,5,7,6]=>2
[[[[[.,.],[.,.]],.],.],[.,.]]=>[1,3,2,4,5,7,6]=>2
[[[[[[.,.],.],.],.],.],[.,.]]=>[1,2,3,4,5,7,6]=>1
[[[.,.],[.,[[.,.],[.,.]]]],.]=>[1,4,6,5,3,2,7]=>8
[[[.,.],[.,[[[.,.],.],.]]],.]=>[1,4,5,6,3,2,7]=>7
[[[.,.],[[.,.],[.,[.,.]]]],.]=>[1,3,6,5,4,2,7]=>7
[[[.,.],[[.,.],[[.,.],.]]],.]=>[1,3,5,6,4,2,7]=>6
[[[.,.],[[.,[.,.]],[.,.]]],.]=>[1,4,3,6,5,2,7]=>6
[[[.,.],[[[.,.],.],[.,.]]],.]=>[1,3,4,6,5,2,7]=>5
[[[.,.],[[.,[[.,.],.]],.]],.]=>[1,4,5,3,6,2,7]=>6
[[[.,.],[[[.,.],[.,.]],.]],.]=>[1,3,5,4,6,2,7]=>5
[[[.,.],[[[.,[.,.]],.],.]],.]=>[1,4,3,5,6,2,7]=>5
[[[.,.],[[[[.,.],.],.],.]],.]=>[1,3,4,5,6,2,7]=>4
[[[[.,.],.],[.,[.,[.,.]]]],.]=>[1,2,6,5,4,3,7]=>6
[[[[.,.],.],[.,[[.,.],.]]],.]=>[1,2,5,6,4,3,7]=>5
[[[[.,.],.],[[.,.],[.,.]]],.]=>[1,2,4,6,5,3,7]=>4
[[[[.,.],.],[[.,[.,.]],.]],.]=>[1,2,5,4,6,3,7]=>4
[[[[.,.],.],[[[.,.],.],.]],.]=>[1,2,4,5,6,3,7]=>3
[[[[.,.],[.,.]],[.,[.,.]]],.]=>[1,3,2,6,5,4,7]=>4
[[[[.,.],[.,.]],[[.,.],.]],.]=>[1,3,2,5,6,4,7]=>3
[[[[[.,.],.],.],[.,[.,.]]],.]=>[1,2,3,6,5,4,7]=>3
[[[[[.,.],.],.],[[.,.],.]],.]=>[1,2,3,5,6,4,7]=>2
[[[[.,.],[.,[.,.]]],[.,.]],.]=>[1,4,3,2,6,5,7]=>4
[[[[.,.],[[.,.],.]],[.,.]],.]=>[1,3,4,2,6,5,7]=>3
[[[[[.,.],.],[.,.]],[.,.]],.]=>[1,2,4,3,6,5,7]=>2
[[[[[.,.],[.,.]],.],[.,.]],.]=>[1,3,2,4,6,5,7]=>2
[[[[[[.,.],.],.],.],[.,.]],.]=>[1,2,3,4,6,5,7]=>1
[[[[.,.],[.,[[.,.],.]]],.],.]=>[1,4,5,3,2,6,7]=>5
[[[[.,.],[[.,.],[.,.]]],.],.]=>[1,3,5,4,2,6,7]=>4
[[[[.,.],[[.,[.,.]],.]],.],.]=>[1,4,3,5,2,6,7]=>4
[[[[.,.],[[[.,.],.],.]],.],.]=>[1,3,4,5,2,6,7]=>3
[[[[[.,.],.],[.,[.,.]]],.],.]=>[1,2,5,4,3,6,7]=>3
[[[[[.,.],.],[[.,.],.]],.],.]=>[1,2,4,5,3,6,7]=>2
[[[[[.,.],[.,.]],[.,.]],.],.]=>[1,3,2,5,4,6,7]=>2
[[[[[[.,.],.],.],[.,.]],.],.]=>[1,2,3,5,4,6,7]=>1
[[[[[.,.],[.,[.,.]]],.],.],.]=>[1,4,3,2,5,6,7]=>3
[[[[[.,.],[[.,.],.]],.],.],.]=>[1,3,4,2,5,6,7]=>2
[[[[[[.,.],.],[.,.]],.],.],.]=>[1,2,4,3,5,6,7]=>1
[[[[[[.,.],[.,.]],.],.],.],.]=>[1,3,2,4,5,6,7]=>1
[[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7]=>0
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Description
The number of transpositions that are smaller or equal to a permutation in Bruhat order.
A statistic is known to be smooth if and only if this number coincides with the number of inversions. This is also equivalent for a permutation to avoid the two pattern $4231$ and $3412$.
A statistic is known to be smooth if and only if this number coincides with the number of inversions. This is also equivalent for a permutation to avoid the two pattern $4231$ and $3412$.
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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