Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
St000654: Permutations ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
[.,[.,.]]=>[2,1]=>1
[[.,.],.]=>[1,2]=>2
[.,[.,[.,.]]]=>[3,2,1]=>1
[.,[[.,.],.]]=>[2,3,1]=>2
[[.,.],[.,.]]=>[1,3,2]=>2
[[.,[.,.]],.]=>[2,1,3]=>1
[[[.,.],.],.]=>[1,2,3]=>3
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>1
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>2
[.,[[.,.],[.,.]]]=>[2,4,3,1]=>2
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>1
[.,[[[.,.],.],.]]=>[2,3,4,1]=>3
[[.,.],[.,[.,.]]]=>[1,4,3,2]=>2
[[.,.],[[.,.],.]]=>[1,3,4,2]=>3
[[.,[.,.]],[.,.]]=>[2,1,4,3]=>1
[[[.,.],.],[.,.]]=>[1,2,4,3]=>3
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>1
[[.,[[.,.],.]],.]=>[2,3,1,4]=>2
[[[.,.],[.,.]],.]=>[1,3,2,4]=>2
[[[.,[.,.]],.],.]=>[2,1,3,4]=>1
[[[[.,.],.],.],.]=>[1,2,3,4]=>4
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>1
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>2
[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>2
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>1
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>3
[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>2
[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>3
[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>1
[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>3
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>1
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>2
[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>2
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>1
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>4
[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>2
[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>3
[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>3
[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>2
[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>4
[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>1
[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>1
[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>3
[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>4
[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>1
[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>2
[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>2
[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>1
[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>4
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>1
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>2
[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>2
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>1
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>3
[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>2
[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>3
[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>1
[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>3
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>1
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>2
[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>2
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>1
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>5
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>1
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>2
[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>2
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>1
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>3
[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>2
[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>3
[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>1
[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>3
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>1
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>2
[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>2
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>1
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>4
[.,[[.,.],[.,[.,[.,.]]]]]=>[2,6,5,4,3,1]=>2
[.,[[.,.],[.,[[.,.],.]]]]=>[2,5,6,4,3,1]=>3
[.,[[.,.],[[.,.],[.,.]]]]=>[2,4,6,5,3,1]=>3
[.,[[.,.],[[.,[.,.]],.]]]=>[2,5,4,6,3,1]=>2
[.,[[.,.],[[[.,.],.],.]]]=>[2,4,5,6,3,1]=>4
[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>1
[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>1
[.,[[[.,.],.],[.,[.,.]]]]=>[2,3,6,5,4,1]=>3
[.,[[[.,.],.],[[.,.],.]]]=>[2,3,5,6,4,1]=>4
[.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>1
[.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>2
[.,[[[.,.],[.,.]],[.,.]]]=>[2,4,3,6,5,1]=>2
[.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>1
[.,[[[[.,.],.],.],[.,.]]]=>[2,3,4,6,5,1]=>4
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>1
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>2
[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>2
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>1
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>3
[.,[[[.,.],[.,[.,.]]],.]]=>[2,5,4,3,6,1]=>2
[.,[[[.,.],[[.,.],.]],.]]=>[2,4,5,3,6,1]=>3
[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>1
[.,[[[[.,.],.],[.,.]],.]]=>[2,3,5,4,6,1]=>3
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>1
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>2
[.,[[[[.,.],[.,.]],.],.]]=>[2,4,3,5,6,1]=>2
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>1
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>5
[[.,.],[.,[.,[.,[.,.]]]]]=>[1,6,5,4,3,2]=>2
[[.,.],[.,[.,[[.,.],.]]]]=>[1,5,6,4,3,2]=>3
[[.,.],[.,[[.,.],[.,.]]]]=>[1,4,6,5,3,2]=>3
[[.,.],[.,[[.,[.,.]],.]]]=>[1,5,4,6,3,2]=>2
[[.,.],[.,[[[.,.],.],.]]]=>[1,4,5,6,3,2]=>4
[[.,.],[[.,.],[.,[.,.]]]]=>[1,3,6,5,4,2]=>3
[[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2]=>4
[[.,.],[[.,[.,.]],[.,.]]]=>[1,4,3,6,5,2]=>2
[[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2]=>4
[[.,.],[[.,[.,[.,.]]],.]]=>[1,5,4,3,6,2]=>2
[[.,.],[[.,[[.,.],.]],.]]=>[1,4,5,3,6,2]=>3
[[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2]=>3
[[.,.],[[[.,[.,.]],.],.]]=>[1,4,3,5,6,2]=>2
[[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2]=>5
[[.,[.,.]],[.,[.,[.,.]]]]=>[2,1,6,5,4,3]=>1
[[.,[.,.]],[.,[[.,.],.]]]=>[2,1,5,6,4,3]=>1
[[.,[.,.]],[[.,.],[.,.]]]=>[2,1,4,6,5,3]=>1
[[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3]=>1
[[.,[.,.]],[[[.,.],.],.]]=>[2,1,4,5,6,3]=>1
[[[.,.],.],[.,[.,[.,.]]]]=>[1,2,6,5,4,3]=>3
[[[.,.],.],[.,[[.,.],.]]]=>[1,2,5,6,4,3]=>4
[[[.,.],.],[[.,.],[.,.]]]=>[1,2,4,6,5,3]=>4
[[[.,.],.],[[.,[.,.]],.]]=>[1,2,5,4,6,3]=>3
[[[.,.],.],[[[.,.],.],.]]=>[1,2,4,5,6,3]=>5
[[.,[.,[.,.]]],[.,[.,.]]]=>[3,2,1,6,5,4]=>1
[[.,[.,[.,.]]],[[.,.],.]]=>[3,2,1,5,6,4]=>1
[[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4]=>2
[[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4]=>2
[[[.,.],[.,.]],[.,[.,.]]]=>[1,3,2,6,5,4]=>2
[[[.,.],[.,.]],[[.,.],.]]=>[1,3,2,5,6,4]=>2
[[[.,[.,.]],.],[.,[.,.]]]=>[2,1,3,6,5,4]=>1
[[[.,[.,.]],.],[[.,.],.]]=>[2,1,3,5,6,4]=>1
[[[[.,.],.],.],[.,[.,.]]]=>[1,2,3,6,5,4]=>4
[[[[.,.],.],.],[[.,.],.]]=>[1,2,3,5,6,4]=>5
[[.,[.,[.,[.,.]]]],[.,.]]=>[4,3,2,1,6,5]=>1
[[.,[.,[[.,.],.]]],[.,.]]=>[3,4,2,1,6,5]=>2
[[.,[[.,.],[.,.]]],[.,.]]=>[2,4,3,1,6,5]=>2
[[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>1
[[.,[[[.,.],.],.]],[.,.]]=>[2,3,4,1,6,5]=>3
[[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5]=>2
[[[.,.],[[.,.],.]],[.,.]]=>[1,3,4,2,6,5]=>3
[[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>1
[[[[.,.],.],[.,.]],[.,.]]=>[1,2,4,3,6,5]=>3
[[[.,[.,[.,.]]],.],[.,.]]=>[3,2,1,4,6,5]=>1
[[[.,[[.,.],.]],.],[.,.]]=>[2,3,1,4,6,5]=>2
[[[[.,.],[.,.]],.],[.,.]]=>[1,3,2,4,6,5]=>2
[[[[.,[.,.]],.],.],[.,.]]=>[2,1,3,4,6,5]=>1
[[[[[.,.],.],.],.],[.,.]]=>[1,2,3,4,6,5]=>5
[[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>1
[[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>2
[[.,[.,[[.,.],[.,.]]]],.]=>[3,5,4,2,1,6]=>2
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>1
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>3
[[.,[[.,.],[.,[.,.]]]],.]=>[2,5,4,3,1,6]=>2
[[.,[[.,.],[[.,.],.]]],.]=>[2,4,5,3,1,6]=>3
[[.,[[.,[.,.]],[.,.]]],.]=>[3,2,5,4,1,6]=>1
[[.,[[[.,.],.],[.,.]]],.]=>[2,3,5,4,1,6]=>3
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>1
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>2
[[.,[[[.,.],[.,.]],.]],.]=>[2,4,3,5,1,6]=>2
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>1
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>4
[[[.,.],[.,[.,[.,.]]]],.]=>[1,5,4,3,2,6]=>2
[[[.,.],[.,[[.,.],.]]],.]=>[1,4,5,3,2,6]=>3
[[[.,.],[[.,.],[.,.]]],.]=>[1,3,5,4,2,6]=>3
[[[.,.],[[.,[.,.]],.]],.]=>[1,4,3,5,2,6]=>2
[[[.,.],[[[.,.],.],.]],.]=>[1,3,4,5,2,6]=>4
[[[.,[.,.]],[.,[.,.]]],.]=>[2,1,5,4,3,6]=>1
[[[.,[.,.]],[[.,.],.]],.]=>[2,1,4,5,3,6]=>1
[[[[.,.],.],[.,[.,.]]],.]=>[1,2,5,4,3,6]=>3
[[[[.,.],.],[[.,.],.]],.]=>[1,2,4,5,3,6]=>4
[[[.,[.,[.,.]]],[.,.]],.]=>[3,2,1,5,4,6]=>1
[[[.,[[.,.],.]],[.,.]],.]=>[2,3,1,5,4,6]=>2
[[[[.,.],[.,.]],[.,.]],.]=>[1,3,2,5,4,6]=>2
[[[[.,[.,.]],.],[.,.]],.]=>[2,1,3,5,4,6]=>1
[[[[[.,.],.],.],[.,.]],.]=>[1,2,3,5,4,6]=>4
[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>1
[[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>2
[[[.,[[.,.],[.,.]]],.],.]=>[2,4,3,1,5,6]=>2
[[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>1
[[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>3
[[[[.,.],[.,[.,.]]],.],.]=>[1,4,3,2,5,6]=>2
[[[[.,.],[[.,.],.]],.],.]=>[1,3,4,2,5,6]=>3
[[[[.,[.,.]],[.,.]],.],.]=>[2,1,4,3,5,6]=>1
[[[[[.,.],.],[.,.]],.],.]=>[1,2,4,3,5,6]=>3
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>1
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>2
[[[[[.,.],[.,.]],.],.],.]=>[1,3,2,4,5,6]=>2
[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>1
[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>6
[[.,.],[.,[[[.,.],.],[.,.]]]]=>[1,4,5,7,6,3,2]=>4
[[.,.],[.,[[[[.,.],.],.],.]]]=>[1,4,5,6,7,3,2]=>5
[[.,.],[[.,.],[.,[.,[.,.]]]]]=>[1,3,7,6,5,4,2]=>3
[[.,.],[[.,.],[.,[[.,.],.]]]]=>[1,3,6,7,5,4,2]=>4
[[.,.],[[.,.],[[.,.],[.,.]]]]=>[1,3,5,7,6,4,2]=>4
[[.,.],[[.,.],[[.,[.,.]],.]]]=>[1,3,6,5,7,4,2]=>3
[[.,.],[[.,.],[[[.,.],.],.]]]=>[1,3,5,6,7,4,2]=>5
[[.,.],[[.,[.,.]],[.,[.,.]]]]=>[1,4,3,7,6,5,2]=>2
[[.,.],[[.,[.,.]],[[.,.],.]]]=>[1,4,3,6,7,5,2]=>2
[[.,.],[[[.,.],.],[.,[.,.]]]]=>[1,3,4,7,6,5,2]=>4
[[.,.],[[[.,.],.],[[.,.],.]]]=>[1,3,4,6,7,5,2]=>5
[[.,.],[[.,[[.,.],.]],[.,.]]]=>[1,4,5,3,7,6,2]=>3
[[.,.],[[[.,.],[.,.]],[.,.]]]=>[1,3,5,4,7,6,2]=>3
[[.,.],[[[.,[.,.]],.],[.,.]]]=>[1,4,3,5,7,6,2]=>2
[[.,.],[[[[.,.],.],.],[.,.]]]=>[1,3,4,5,7,6,2]=>5
[[.,.],[[.,[[.,.],[.,.]]],.]]=>[1,4,6,5,3,7,2]=>3
[[.,.],[[.,[[[.,.],.],.]],.]]=>[1,4,5,6,3,7,2]=>4
[[.,.],[[[.,.],[.,[.,.]]],.]]=>[1,3,6,5,4,7,2]=>3
[[.,.],[[[.,.],[[.,.],.]],.]]=>[1,3,5,6,4,7,2]=>4
[[.,.],[[[.,[.,.]],[.,.]],.]]=>[1,4,3,6,5,7,2]=>2
[[.,.],[[[[.,.],.],[.,.]],.]]=>[1,3,4,6,5,7,2]=>4
[[.,.],[[[.,[[.,.],.]],.],.]]=>[1,4,5,3,6,7,2]=>3
[[.,.],[[[[.,.],[.,.]],.],.]]=>[1,3,5,4,6,7,2]=>3
[[.,.],[[[[.,[.,.]],.],.],.]]=>[1,4,3,5,6,7,2]=>2
[[.,.],[[[[[.,.],.],.],.],.]]=>[1,3,4,5,6,7,2]=>6
[[[.,.],.],[.,[.,[.,[.,.]]]]]=>[1,2,7,6,5,4,3]=>3
[[[.,.],.],[.,[.,[[.,.],.]]]]=>[1,2,6,7,5,4,3]=>4
[[[.,.],.],[.,[[.,.],[.,.]]]]=>[1,2,5,7,6,4,3]=>4
[[[.,.],.],[.,[[.,[.,.]],.]]]=>[1,2,6,5,7,4,3]=>3
[[[.,.],.],[.,[[[.,.],.],.]]]=>[1,2,5,6,7,4,3]=>5
[[[.,.],.],[[.,.],[.,[.,.]]]]=>[1,2,4,7,6,5,3]=>4
[[[.,.],.],[[.,.],[[.,.],.]]]=>[1,2,4,6,7,5,3]=>5
[[[.,.],.],[[.,[.,.]],[.,.]]]=>[1,2,5,4,7,6,3]=>3
[[[.,.],.],[[[.,.],.],[.,.]]]=>[1,2,4,5,7,6,3]=>5
[[[.,.],.],[[.,[.,[.,.]]],.]]=>[1,2,6,5,4,7,3]=>3
[[[.,.],.],[[.,[[.,.],.]],.]]=>[1,2,5,6,4,7,3]=>4
[[[.,.],.],[[[.,.],[.,.]],.]]=>[1,2,4,6,5,7,3]=>4
[[[.,.],.],[[[.,[.,.]],.],.]]=>[1,2,5,4,6,7,3]=>3
[[[.,.],.],[[[[.,.],.],.],.]]=>[1,2,4,5,6,7,3]=>6
[[[.,.],[.,.]],[.,[.,[.,.]]]]=>[1,3,2,7,6,5,4]=>2
[[[.,.],[.,.]],[.,[[.,.],.]]]=>[1,3,2,6,7,5,4]=>2
[[[.,.],[.,.]],[[.,.],[.,.]]]=>[1,3,2,5,7,6,4]=>2
[[[.,.],[.,.]],[[.,[.,.]],.]]=>[1,3,2,6,5,7,4]=>2
[[[.,.],[.,.]],[[[.,.],.],.]]=>[1,3,2,5,6,7,4]=>2
[[[[.,.],.],.],[.,[.,[.,.]]]]=>[1,2,3,7,6,5,4]=>4
[[[[.,.],.],.],[.,[[.,.],.]]]=>[1,2,3,6,7,5,4]=>5
[[[[.,.],.],.],[[.,.],[.,.]]]=>[1,2,3,5,7,6,4]=>5
[[[[.,.],.],.],[[.,[.,.]],.]]=>[1,2,3,6,5,7,4]=>4
[[[[.,.],.],.],[[[.,.],.],.]]=>[1,2,3,5,6,7,4]=>6
[[[.,.],[.,[.,.]]],[.,[.,.]]]=>[1,4,3,2,7,6,5]=>2
[[[.,.],[.,[.,.]]],[[.,.],.]]=>[1,4,3,2,6,7,5]=>2
[[[.,.],[[.,.],.]],[.,[.,.]]]=>[1,3,4,2,7,6,5]=>3
[[[.,.],[[.,.],.]],[[.,.],.]]=>[1,3,4,2,6,7,5]=>3
[[[[.,.],.],[.,.]],[.,[.,.]]]=>[1,2,4,3,7,6,5]=>3
[[[[.,.],.],[.,.]],[[.,.],.]]=>[1,2,4,3,6,7,5]=>3
[[[[.,.],[.,.]],.],[.,[.,.]]]=>[1,3,2,4,7,6,5]=>2
[[[[.,.],[.,.]],.],[[.,.],.]]=>[1,3,2,4,6,7,5]=>2
[[[[[.,.],.],.],.],[.,[.,.]]]=>[1,2,3,4,7,6,5]=>5
[[[[[.,.],.],.],.],[[.,.],.]]=>[1,2,3,4,6,7,5]=>6
[[[.,.],[.,[[.,.],.]]],[.,.]]=>[1,4,5,3,2,7,6]=>3
[[[.,.],[[.,.],[.,.]]],[.,.]]=>[1,3,5,4,2,7,6]=>3
[[[.,.],[[.,[.,.]],.]],[.,.]]=>[1,4,3,5,2,7,6]=>2
[[[.,.],[[[.,.],.],.]],[.,.]]=>[1,3,4,5,2,7,6]=>4
[[[[.,.],.],[.,[.,.]]],[.,.]]=>[1,2,5,4,3,7,6]=>3
[[[[.,.],.],[[.,.],.]],[.,.]]=>[1,2,4,5,3,7,6]=>4
[[[[.,.],[.,.]],[.,.]],[.,.]]=>[1,3,2,5,4,7,6]=>2
[[[[[.,.],.],.],[.,.]],[.,.]]=>[1,2,3,5,4,7,6]=>4
[[[[.,.],[.,[.,.]]],.],[.,.]]=>[1,4,3,2,5,7,6]=>2
[[[[.,.],[[.,.],.]],.],[.,.]]=>[1,3,4,2,5,7,6]=>3
[[[[[.,.],.],[.,.]],.],[.,.]]=>[1,2,4,3,5,7,6]=>3
[[[[[.,.],[.,.]],.],.],[.,.]]=>[1,3,2,4,5,7,6]=>2
[[[[[[.,.],.],.],.],.],[.,.]]=>[1,2,3,4,5,7,6]=>6
[[[.,.],[.,[[.,.],[.,.]]]],.]=>[1,4,6,5,3,2,7]=>3
[[[.,.],[.,[[[.,.],.],.]]],.]=>[1,4,5,6,3,2,7]=>4
[[[.,.],[[.,.],[.,[.,.]]]],.]=>[1,3,6,5,4,2,7]=>3
[[[.,.],[[.,.],[[.,.],.]]],.]=>[1,3,5,6,4,2,7]=>4
[[[.,.],[[.,[.,.]],[.,.]]],.]=>[1,4,3,6,5,2,7]=>2
[[[.,.],[[[.,.],.],[.,.]]],.]=>[1,3,4,6,5,2,7]=>4
[[[.,.],[[.,[[.,.],.]],.]],.]=>[1,4,5,3,6,2,7]=>3
[[[.,.],[[[.,.],[.,.]],.]],.]=>[1,3,5,4,6,2,7]=>3
[[[.,.],[[[.,[.,.]],.],.]],.]=>[1,4,3,5,6,2,7]=>2
[[[.,.],[[[[.,.],.],.],.]],.]=>[1,3,4,5,6,2,7]=>5
[[[[.,.],.],[.,[.,[.,.]]]],.]=>[1,2,6,5,4,3,7]=>3
[[[[.,.],.],[.,[[.,.],.]]],.]=>[1,2,5,6,4,3,7]=>4
[[[[.,.],.],[[.,.],[.,.]]],.]=>[1,2,4,6,5,3,7]=>4
[[[[.,.],.],[[.,[.,.]],.]],.]=>[1,2,5,4,6,3,7]=>3
[[[[.,.],.],[[[.,.],.],.]],.]=>[1,2,4,5,6,3,7]=>5
[[[[.,.],[.,.]],[.,[.,.]]],.]=>[1,3,2,6,5,4,7]=>2
[[[[.,.],[.,.]],[[.,.],.]],.]=>[1,3,2,5,6,4,7]=>2
[[[[[.,.],.],.],[.,[.,.]]],.]=>[1,2,3,6,5,4,7]=>4
[[[[[.,.],.],.],[[.,.],.]],.]=>[1,2,3,5,6,4,7]=>5
[[[[.,.],[.,[.,.]]],[.,.]],.]=>[1,4,3,2,6,5,7]=>2
[[[[.,.],[[.,.],.]],[.,.]],.]=>[1,3,4,2,6,5,7]=>3
[[[[[.,.],.],[.,.]],[.,.]],.]=>[1,2,4,3,6,5,7]=>3
[[[[[.,.],[.,.]],.],[.,.]],.]=>[1,3,2,4,6,5,7]=>2
[[[[[[.,.],.],.],.],[.,.]],.]=>[1,2,3,4,6,5,7]=>5
[[[[.,.],[.,[[.,.],.]]],.],.]=>[1,4,5,3,2,6,7]=>3
[[[[.,.],[[.,.],[.,.]]],.],.]=>[1,3,5,4,2,6,7]=>3
[[[[.,.],[[.,[.,.]],.]],.],.]=>[1,4,3,5,2,6,7]=>2
[[[[.,.],[[[.,.],.],.]],.],.]=>[1,3,4,5,2,6,7]=>4
[[[[[.,.],.],[.,[.,.]]],.],.]=>[1,2,5,4,3,6,7]=>3
[[[[[.,.],.],[[.,.],.]],.],.]=>[1,2,4,5,3,6,7]=>4
[[[[[.,.],[.,.]],[.,.]],.],.]=>[1,3,2,5,4,6,7]=>2
[[[[[[.,.],.],.],[.,.]],.],.]=>[1,2,3,5,4,6,7]=>4
[[[[[.,.],[.,[.,.]]],.],.],.]=>[1,4,3,2,5,6,7]=>2
[[[[[.,.],[[.,.],.]],.],.],.]=>[1,3,4,2,5,6,7]=>3
[[[[[[.,.],.],[.,.]],.],.],.]=>[1,2,4,3,5,6,7]=>3
[[[[[[.,.],[.,.]],.],.],.],.]=>[1,3,2,4,5,6,7]=>2
[[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7]=>7
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Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
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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