Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
St000829: Permutations ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
[.,[.,.]]=>[2,1]=>1
[[.,.],.]=>[1,2]=>0
[.,[.,[.,.]]]=>[3,2,1]=>2
[.,[[.,.],.]]=>[2,3,1]=>1
[[.,.],[.,.]]=>[1,3,2]=>1
[[.,[.,.]],.]=>[2,1,3]=>1
[[[.,.],.],.]=>[1,2,3]=>0
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>3
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>2
[.,[[.,.],[.,.]]]=>[2,4,3,1]=>2
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>2
[.,[[[.,.],.],.]]=>[2,3,4,1]=>1
[[.,.],[.,[.,.]]]=>[1,4,3,2]=>2
[[.,.],[[.,.],.]]=>[1,3,4,2]=>1
[[.,[.,.]],[.,.]]=>[2,1,4,3]=>2
[[[.,.],.],[.,.]]=>[1,2,4,3]=>1
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>2
[[.,[[.,.],.]],.]=>[2,3,1,4]=>1
[[[.,.],[.,.]],.]=>[1,3,2,4]=>1
[[[.,[.,.]],.],.]=>[2,1,3,4]=>1
[[[[.,.],.],.],.]=>[1,2,3,4]=>0
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>4
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>3
[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>3
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>3
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>2
[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>3
[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>2
[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>3
[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>2
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>3
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>2
[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>2
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>2
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>1
[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>3
[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>2
[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>2
[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>2
[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>1
[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>3
[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>2
[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>2
[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>1
[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>3
[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>2
[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>2
[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>2
[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>1
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>3
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>2
[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>2
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>2
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>1
[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>2
[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>1
[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>2
[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>1
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>2
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>1
[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>1
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>1
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>0
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>5
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>4
[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>4
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>4
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>3
[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>4
[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>3
[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>4
[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>3
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>4
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>3
[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>3
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>3
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>2
[.,[[.,.],[.,[.,[.,.]]]]]=>[2,6,5,4,3,1]=>4
[.,[[.,.],[.,[[.,.],.]]]]=>[2,5,6,4,3,1]=>3
[.,[[.,.],[[.,.],[.,.]]]]=>[2,4,6,5,3,1]=>3
[.,[[.,.],[[.,[.,.]],.]]]=>[2,5,4,6,3,1]=>3
[.,[[.,.],[[[.,.],.],.]]]=>[2,4,5,6,3,1]=>2
[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>4
[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>3
[.,[[[.,.],.],[.,[.,.]]]]=>[2,3,6,5,4,1]=>3
[.,[[[.,.],.],[[.,.],.]]]=>[2,3,5,6,4,1]=>2
[.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>4
[.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>3
[.,[[[.,.],[.,.]],[.,.]]]=>[2,4,3,6,5,1]=>3
[.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>3
[.,[[[[.,.],.],.],[.,.]]]=>[2,3,4,6,5,1]=>2
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>4
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>3
[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>3
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>3
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>2
[.,[[[.,.],[.,[.,.]]],.]]=>[2,5,4,3,6,1]=>3
[.,[[[.,.],[[.,.],.]],.]]=>[2,4,5,3,6,1]=>2
[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>3
[.,[[[[.,.],.],[.,.]],.]]=>[2,3,5,4,6,1]=>2
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>3
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>2
[.,[[[[.,.],[.,.]],.],.]]=>[2,4,3,5,6,1]=>2
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>2
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>1
[[.,.],[.,[.,[.,[.,.]]]]]=>[1,6,5,4,3,2]=>4
[[.,.],[.,[.,[[.,.],.]]]]=>[1,5,6,4,3,2]=>3
[[.,.],[.,[[.,.],[.,.]]]]=>[1,4,6,5,3,2]=>3
[[.,.],[.,[[.,[.,.]],.]]]=>[1,5,4,6,3,2]=>3
[[.,.],[.,[[[.,.],.],.]]]=>[1,4,5,6,3,2]=>2
[[.,.],[[.,.],[.,[.,.]]]]=>[1,3,6,5,4,2]=>3
[[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2]=>2
[[.,.],[[.,[.,.]],[.,.]]]=>[1,4,3,6,5,2]=>3
[[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2]=>2
[[.,.],[[.,[.,[.,.]]],.]]=>[1,5,4,3,6,2]=>3
[[.,.],[[.,[[.,.],.]],.]]=>[1,4,5,3,6,2]=>2
[[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2]=>2
[[.,.],[[[.,[.,.]],.],.]]=>[1,4,3,5,6,2]=>2
[[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2]=>1
[[.,[.,.]],[.,[.,[.,.]]]]=>[2,1,6,5,4,3]=>4
[[.,[.,.]],[.,[[.,.],.]]]=>[2,1,5,6,4,3]=>3
[[.,[.,.]],[[.,.],[.,.]]]=>[2,1,4,6,5,3]=>3
[[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3]=>3
[[.,[.,.]],[[[.,.],.],.]]=>[2,1,4,5,6,3]=>2
[[[.,.],.],[.,[.,[.,.]]]]=>[1,2,6,5,4,3]=>3
[[[.,.],.],[.,[[.,.],.]]]=>[1,2,5,6,4,3]=>2
[[[.,.],.],[[.,.],[.,.]]]=>[1,2,4,6,5,3]=>2
[[[.,.],.],[[.,[.,.]],.]]=>[1,2,5,4,6,3]=>2
[[[.,.],.],[[[.,.],.],.]]=>[1,2,4,5,6,3]=>1
[[.,[.,[.,.]]],[.,[.,.]]]=>[3,2,1,6,5,4]=>4
[[.,[.,[.,.]]],[[.,.],.]]=>[3,2,1,5,6,4]=>3
[[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4]=>3
[[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4]=>2
[[[.,.],[.,.]],[.,[.,.]]]=>[1,3,2,6,5,4]=>3
[[[.,.],[.,.]],[[.,.],.]]=>[1,3,2,5,6,4]=>2
[[[.,[.,.]],.],[.,[.,.]]]=>[2,1,3,6,5,4]=>3
[[[.,[.,.]],.],[[.,.],.]]=>[2,1,3,5,6,4]=>2
[[[[.,.],.],.],[.,[.,.]]]=>[1,2,3,6,5,4]=>2
[[[[.,.],.],.],[[.,.],.]]=>[1,2,3,5,6,4]=>1
[[.,[.,[.,[.,.]]]],[.,.]]=>[4,3,2,1,6,5]=>4
[[.,[.,[[.,.],.]]],[.,.]]=>[3,4,2,1,6,5]=>3
[[.,[[.,.],[.,.]]],[.,.]]=>[2,4,3,1,6,5]=>3
[[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>3
[[.,[[[.,.],.],.]],[.,.]]=>[2,3,4,1,6,5]=>2
[[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5]=>3
[[[.,.],[[.,.],.]],[.,.]]=>[1,3,4,2,6,5]=>2
[[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>3
[[[[.,.],.],[.,.]],[.,.]]=>[1,2,4,3,6,5]=>2
[[[.,[.,[.,.]]],.],[.,.]]=>[3,2,1,4,6,5]=>3
[[[.,[[.,.],.]],.],[.,.]]=>[2,3,1,4,6,5]=>2
[[[[.,.],[.,.]],.],[.,.]]=>[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]=>4
[[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>3
[[.,[.,[[.,.],[.,.]]]],.]=>[3,5,4,2,1,6]=>3
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>3
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>2
[[.,[[.,.],[.,[.,.]]]],.]=>[2,5,4,3,1,6]=>3
[[.,[[.,.],[[.,.],.]]],.]=>[2,4,5,3,1,6]=>2
[[.,[[.,[.,.]],[.,.]]],.]=>[3,2,5,4,1,6]=>3
[[.,[[[.,.],.],[.,.]]],.]=>[2,3,5,4,1,6]=>2
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>3
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>2
[[.,[[[.,.],[.,.]],.]],.]=>[2,4,3,5,1,6]=>2
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>2
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>1
[[[.,.],[.,[.,[.,.]]]],.]=>[1,5,4,3,2,6]=>3
[[[.,.],[.,[[.,.],.]]],.]=>[1,4,5,3,2,6]=>2
[[[.,.],[[.,.],[.,.]]],.]=>[1,3,5,4,2,6]=>2
[[[.,.],[[.,[.,.]],.]],.]=>[1,4,3,5,2,6]=>2
[[[.,.],[[[.,.],.],.]],.]=>[1,3,4,5,2,6]=>1
[[[.,[.,.]],[.,[.,.]]],.]=>[2,1,5,4,3,6]=>3
[[[.,[.,.]],[[.,.],.]],.]=>[2,1,4,5,3,6]=>2
[[[[.,.],.],[.,[.,.]]],.]=>[1,2,5,4,3,6]=>2
[[[[.,.],.],[[.,.],.]],.]=>[1,2,4,5,3,6]=>1
[[[.,[.,[.,.]]],[.,.]],.]=>[3,2,1,5,4,6]=>3
[[[.,[[.,.],.]],[.,.]],.]=>[2,3,1,5,4,6]=>2
[[[[.,.],[.,.]],[.,.]],.]=>[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]=>3
[[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>2
[[[.,[[.,.],[.,.]]],.],.]=>[2,4,3,1,5,6]=>2
[[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>2
[[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>1
[[[[.,.],[.,[.,.]]],.],.]=>[1,4,3,2,5,6]=>2
[[[[.,.],[[.,.],.]],.],.]=>[1,3,4,2,5,6]=>1
[[[[.,[.,.]],[.,.]],.],.]=>[2,1,4,3,5,6]=>2
[[[[[.,.],.],[.,.]],.],.]=>[1,2,4,3,5,6]=>1
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>2
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>1
[[[[[.,.],[.,.]],.],.],.]=>[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]=>3
[[.,.],[.,[[[[.,.],.],.],.]]]=>[1,4,5,6,7,3,2]=>2
[[.,.],[[.,.],[.,[.,[.,.]]]]]=>[1,3,7,6,5,4,2]=>4
[[.,.],[[.,.],[.,[[.,.],.]]]]=>[1,3,6,7,5,4,2]=>3
[[.,.],[[.,.],[[.,.],[.,.]]]]=>[1,3,5,7,6,4,2]=>3
[[.,.],[[.,.],[[.,[.,.]],.]]]=>[1,3,6,5,7,4,2]=>3
[[.,.],[[.,.],[[[.,.],.],.]]]=>[1,3,5,6,7,4,2]=>2
[[.,.],[[.,[.,.]],[.,[.,.]]]]=>[1,4,3,7,6,5,2]=>4
[[.,.],[[.,[.,.]],[[.,.],.]]]=>[1,4,3,6,7,5,2]=>3
[[.,.],[[[.,.],.],[.,[.,.]]]]=>[1,3,4,7,6,5,2]=>3
[[.,.],[[[.,.],.],[[.,.],.]]]=>[1,3,4,6,7,5,2]=>2
[[.,.],[[.,[[.,.],.]],[.,.]]]=>[1,4,5,3,7,6,2]=>3
[[.,.],[[[.,.],[.,.]],[.,.]]]=>[1,3,5,4,7,6,2]=>3
[[.,.],[[[.,[.,.]],.],[.,.]]]=>[1,4,3,5,7,6,2]=>3
[[.,.],[[[[.,.],.],.],[.,.]]]=>[1,3,4,5,7,6,2]=>2
[[.,.],[[.,[[.,.],[.,.]]],.]]=>[1,4,6,5,3,7,2]=>3
[[.,.],[[.,[[[.,.],.],.]],.]]=>[1,4,5,6,3,7,2]=>2
[[.,.],[[[.,.],[.,[.,.]]],.]]=>[1,3,6,5,4,7,2]=>3
[[.,.],[[[.,.],[[.,.],.]],.]]=>[1,3,5,6,4,7,2]=>2
[[.,.],[[[.,[.,.]],[.,.]],.]]=>[1,4,3,6,5,7,2]=>3
[[.,.],[[[[.,.],.],[.,.]],.]]=>[1,3,4,6,5,7,2]=>2
[[.,.],[[[.,[[.,.],.]],.],.]]=>[1,4,5,3,6,7,2]=>2
[[.,.],[[[[.,.],[.,.]],.],.]]=>[1,3,5,4,6,7,2]=>2
[[.,.],[[[[.,[.,.]],.],.],.]]=>[1,4,3,5,6,7,2]=>2
[[.,.],[[[[[.,.],.],.],.],.]]=>[1,3,4,5,6,7,2]=>1
[[[.,.],.],[.,[.,[.,[.,.]]]]]=>[1,2,7,6,5,4,3]=>4
[[[.,.],.],[.,[.,[[.,.],.]]]]=>[1,2,6,7,5,4,3]=>3
[[[.,.],.],[.,[[.,.],[.,.]]]]=>[1,2,5,7,6,4,3]=>3
[[[.,.],.],[.,[[.,[.,.]],.]]]=>[1,2,6,5,7,4,3]=>3
[[[.,.],.],[.,[[[.,.],.],.]]]=>[1,2,5,6,7,4,3]=>2
[[[.,.],.],[[.,.],[.,[.,.]]]]=>[1,2,4,7,6,5,3]=>3
[[[.,.],.],[[.,.],[[.,.],.]]]=>[1,2,4,6,7,5,3]=>2
[[[.,.],.],[[.,[.,.]],[.,.]]]=>[1,2,5,4,7,6,3]=>3
[[[.,.],.],[[[.,.],.],[.,.]]]=>[1,2,4,5,7,6,3]=>2
[[[.,.],.],[[.,[.,[.,.]]],.]]=>[1,2,6,5,4,7,3]=>3
[[[.,.],.],[[.,[[.,.],.]],.]]=>[1,2,5,6,4,7,3]=>2
[[[.,.],.],[[[.,.],[.,.]],.]]=>[1,2,4,6,5,7,3]=>2
[[[.,.],.],[[[.,[.,.]],.],.]]=>[1,2,5,4,6,7,3]=>2
[[[.,.],.],[[[[.,.],.],.],.]]=>[1,2,4,5,6,7,3]=>1
[[[.,.],[.,.]],[.,[.,[.,.]]]]=>[1,3,2,7,6,5,4]=>4
[[[.,.],[.,.]],[.,[[.,.],.]]]=>[1,3,2,6,7,5,4]=>3
[[[.,.],[.,.]],[[.,.],[.,.]]]=>[1,3,2,5,7,6,4]=>3
[[[.,.],[.,.]],[[.,[.,.]],.]]=>[1,3,2,6,5,7,4]=>3
[[[.,.],[.,.]],[[[.,.],.],.]]=>[1,3,2,5,6,7,4]=>2
[[[[.,.],.],.],[.,[.,[.,.]]]]=>[1,2,3,7,6,5,4]=>3
[[[[.,.],.],.],[.,[[.,.],.]]]=>[1,2,3,6,7,5,4]=>2
[[[[.,.],.],.],[[.,.],[.,.]]]=>[1,2,3,5,7,6,4]=>2
[[[[.,.],.],.],[[.,[.,.]],.]]=>[1,2,3,6,5,7,4]=>2
[[[[.,.],.],.],[[[.,.],.],.]]=>[1,2,3,5,6,7,4]=>1
[[[.,.],[.,[.,.]]],[.,[.,.]]]=>[1,4,3,2,7,6,5]=>4
[[[.,.],[.,[.,.]]],[[.,.],.]]=>[1,4,3,2,6,7,5]=>3
[[[.,.],[[.,.],.]],[.,[.,.]]]=>[1,3,4,2,7,6,5]=>3
[[[.,.],[[.,.],.]],[[.,.],.]]=>[1,3,4,2,6,7,5]=>2
[[[[.,.],.],[.,.]],[.,[.,.]]]=>[1,2,4,3,7,6,5]=>3
[[[[.,.],.],[.,.]],[[.,.],.]]=>[1,2,4,3,6,7,5]=>2
[[[[.,.],[.,.]],.],[.,[.,.]]]=>[1,3,2,4,7,6,5]=>3
[[[[.,.],[.,.]],.],[[.,.],.]]=>[1,3,2,4,6,7,5]=>2
[[[[[.,.],.],.],.],[.,[.,.]]]=>[1,2,3,4,7,6,5]=>2
[[[[[.,.],.],.],.],[[.,.],.]]=>[1,2,3,4,6,7,5]=>1
[[[.,.],[.,[[.,.],.]]],[.,.]]=>[1,4,5,3,2,7,6]=>3
[[[.,.],[[.,.],[.,.]]],[.,.]]=>[1,3,5,4,2,7,6]=>3
[[[.,.],[[.,[.,.]],.]],[.,.]]=>[1,4,3,5,2,7,6]=>3
[[[.,.],[[[.,.],.],.]],[.,.]]=>[1,3,4,5,2,7,6]=>2
[[[[.,.],.],[.,[.,.]]],[.,.]]=>[1,2,5,4,3,7,6]=>3
[[[[.,.],.],[[.,.],.]],[.,.]]=>[1,2,4,5,3,7,6]=>2
[[[[.,.],[.,.]],[.,.]],[.,.]]=>[1,3,2,5,4,7,6]=>3
[[[[[.,.],.],.],[.,.]],[.,.]]=>[1,2,3,5,4,7,6]=>2
[[[[.,.],[.,[.,.]]],.],[.,.]]=>[1,4,3,2,5,7,6]=>3
[[[[.,.],[[.,.],.]],.],[.,.]]=>[1,3,4,2,5,7,6]=>2
[[[[[.,.],.],[.,.]],.],[.,.]]=>[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]=>3
[[[.,.],[.,[[[.,.],.],.]]],.]=>[1,4,5,6,3,2,7]=>2
[[[.,.],[[.,.],[.,[.,.]]]],.]=>[1,3,6,5,4,2,7]=>3
[[[.,.],[[.,.],[[.,.],.]]],.]=>[1,3,5,6,4,2,7]=>2
[[[.,.],[[.,[.,.]],[.,.]]],.]=>[1,4,3,6,5,2,7]=>3
[[[.,.],[[[.,.],.],[.,.]]],.]=>[1,3,4,6,5,2,7]=>2
[[[.,.],[[.,[[.,.],.]],.]],.]=>[1,4,5,3,6,2,7]=>2
[[[.,.],[[[.,.],[.,.]],.]],.]=>[1,3,5,4,6,2,7]=>2
[[[.,.],[[[.,[.,.]],.],.]],.]=>[1,4,3,5,6,2,7]=>2
[[[.,.],[[[[.,.],.],.],.]],.]=>[1,3,4,5,6,2,7]=>1
[[[[.,.],.],[.,[.,[.,.]]]],.]=>[1,2,6,5,4,3,7]=>3
[[[[.,.],.],[.,[[.,.],.]]],.]=>[1,2,5,6,4,3,7]=>2
[[[[.,.],.],[[.,.],[.,.]]],.]=>[1,2,4,6,5,3,7]=>2
[[[[.,.],.],[[.,[.,.]],.]],.]=>[1,2,5,4,6,3,7]=>2
[[[[.,.],.],[[[.,.],.],.]],.]=>[1,2,4,5,6,3,7]=>1
[[[[.,.],[.,.]],[.,[.,.]]],.]=>[1,3,2,6,5,4,7]=>3
[[[[.,.],[.,.]],[[.,.],.]],.]=>[1,3,2,5,6,4,7]=>2
[[[[[.,.],.],.],[.,[.,.]]],.]=>[1,2,3,6,5,4,7]=>2
[[[[[.,.],.],.],[[.,.],.]],.]=>[1,2,3,5,6,4,7]=>1
[[[[.,.],[.,[.,.]]],[.,.]],.]=>[1,4,3,2,6,5,7]=>3
[[[[.,.],[[.,.],.]],[.,.]],.]=>[1,3,4,2,6,5,7]=>2
[[[[[.,.],.],[.,.]],[.,.]],.]=>[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]=>2
[[[[.,.],[[.,.],[.,.]]],.],.]=>[1,3,5,4,2,6,7]=>2
[[[[.,.],[[.,[.,.]],.]],.],.]=>[1,4,3,5,2,6,7]=>2
[[[[.,.],[[[.,.],.],.]],.],.]=>[1,3,4,5,2,6,7]=>1
[[[[[.,.],.],[.,[.,.]]],.],.]=>[1,2,5,4,3,6,7]=>2
[[[[[.,.],.],[[.,.],.]],.],.]=>[1,2,4,5,3,6,7]=>1
[[[[[.,.],[.,.]],[.,.]],.],.]=>[1,3,2,5,4,6,7]=>2
[[[[[[.,.],.],.],[.,.]],.],.]=>[1,2,3,5,4,6,7]=>1
[[[[[.,.],[.,[.,.]]],.],.],.]=>[1,4,3,2,5,6,7]=>2
[[[[[.,.],[[.,.],.]],.],.],.]=>[1,3,4,2,5,6,7]=>1
[[[[[[.,.],.],[.,.]],.],.],.]=>[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 Ulam distance of a permutation to the identity permutation.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus St000062The length of the longest increasing subsequence of the permutation. equals $n$.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus St000062The length of the longest increasing subsequence of the permutation. equals $n$.
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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