Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000501: Permutations ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
[.,.]=>[1]=>[1]=>1
[.,[.,.]]=>[2,1]=>[1,2]=>1
[[.,.],.]=>[1,2]=>[2,1]=>2
[.,[.,[.,.]]]=>[3,2,1]=>[1,3,2]=>1
[.,[[.,.],.]]=>[2,3,1]=>[1,2,3]=>1
[[.,.],[.,.]]=>[1,3,2]=>[2,1,3]=>2
[[.,[.,.]],.]=>[2,1,3]=>[3,2,1]=>3
[[[.,.],.],.]=>[1,2,3]=>[2,3,1]=>3
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>[1,4,3,2]=>1
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>[1,4,2,3]=>1
[.,[[.,.],[.,.]]]=>[2,4,3,1]=>[1,2,4,3]=>1
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>[1,3,2,4]=>1
[.,[[[.,.],.],.]]=>[2,3,4,1]=>[1,2,3,4]=>1
[[.,.],[.,[.,.]]]=>[1,4,3,2]=>[2,1,4,3]=>2
[[.,.],[[.,.],.]]=>[1,3,4,2]=>[2,1,3,4]=>2
[[.,[.,.]],[.,.]]=>[2,1,4,3]=>[3,2,1,4]=>3
[[[.,.],.],[.,.]]=>[1,2,4,3]=>[2,3,1,4]=>3
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>[4,3,2,1]=>4
[[.,[[.,.],.]],.]=>[2,3,1,4]=>[4,2,3,1]=>4
[[[.,.],[.,.]],.]=>[1,3,2,4]=>[2,4,3,1]=>4
[[[.,[.,.]],.],.]=>[2,1,3,4]=>[3,2,4,1]=>4
[[[[.,.],.],.],.]=>[1,2,3,4]=>[2,3,4,1]=>4
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>[1,5,4,3,2]=>1
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>[1,5,4,2,3]=>1
[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>[1,5,2,4,3]=>1
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>[1,5,3,2,4]=>1
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>[1,5,2,3,4]=>1
[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>[1,2,5,4,3]=>1
[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>[1,2,5,3,4]=>1
[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>[1,3,2,5,4]=>1
[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>[1,2,3,5,4]=>1
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>[1,4,3,2,5]=>1
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>[1,4,2,3,5]=>1
[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>[1,2,4,3,5]=>1
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>[1,3,2,4,5]=>1
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>[1,2,3,4,5]=>1
[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>[2,1,5,4,3]=>2
[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>[2,1,5,3,4]=>2
[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>[2,1,3,5,4]=>2
[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>[2,1,4,3,5]=>2
[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>[2,1,3,4,5]=>2
[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>[3,2,1,5,4]=>3
[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>[3,2,1,4,5]=>3
[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>[2,3,1,5,4]=>3
[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>[2,3,1,4,5]=>3
[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>[4,3,2,1,5]=>4
[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>[4,2,3,1,5]=>4
[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>[2,4,3,1,5]=>4
[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>[3,2,4,1,5]=>4
[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>[2,3,4,1,5]=>4
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>[5,4,3,2,1]=>5
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>[5,4,2,3,1]=>5
[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>[5,2,4,3,1]=>5
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>[5,3,2,4,1]=>5
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>[5,2,3,4,1]=>5
[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>[2,5,4,3,1]=>5
[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>[2,5,3,4,1]=>5
[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>[3,2,5,4,1]=>5
[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>[2,3,5,4,1]=>5
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>[4,3,2,5,1]=>5
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>[4,2,3,5,1]=>5
[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>[2,4,3,5,1]=>5
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>[3,2,4,5,1]=>5
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>[2,3,4,5,1]=>5
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>[1,6,5,4,3,2]=>1
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>[1,6,5,4,2,3]=>1
[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>[1,6,5,2,4,3]=>1
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>[1,6,5,3,2,4]=>1
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>[1,6,5,2,3,4]=>1
[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>[1,6,2,5,4,3]=>1
[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>[1,6,2,5,3,4]=>1
[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>[1,6,3,2,5,4]=>1
[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>[1,6,2,3,5,4]=>1
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>[1,6,4,3,2,5]=>1
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>[1,6,4,2,3,5]=>1
[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>[1,6,2,4,3,5]=>1
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>[1,6,3,2,4,5]=>1
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>[1,6,2,3,4,5]=>1
[.,[[.,.],[.,[.,[.,.]]]]]=>[2,6,5,4,3,1]=>[1,2,6,5,4,3]=>1
[.,[[.,.],[.,[[.,.],.]]]]=>[2,5,6,4,3,1]=>[1,2,6,5,3,4]=>1
[.,[[.,.],[[.,.],[.,.]]]]=>[2,4,6,5,3,1]=>[1,2,6,3,5,4]=>1
[.,[[.,.],[[.,[.,.]],.]]]=>[2,5,4,6,3,1]=>[1,2,6,4,3,5]=>1
[.,[[.,.],[[[.,.],.],.]]]=>[2,4,5,6,3,1]=>[1,2,6,3,4,5]=>1
[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>[1,3,2,6,5,4]=>1
[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>[1,3,2,6,4,5]=>1
[.,[[[.,.],.],[.,[.,.]]]]=>[2,3,6,5,4,1]=>[1,2,3,6,5,4]=>1
[.,[[[.,.],.],[[.,.],.]]]=>[2,3,5,6,4,1]=>[1,2,3,6,4,5]=>1
[.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>[1,4,3,2,6,5]=>1
[.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>[1,4,2,3,6,5]=>1
[.,[[[.,.],[.,.]],[.,.]]]=>[2,4,3,6,5,1]=>[1,2,4,3,6,5]=>1
[.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>[1,3,2,4,6,5]=>1
[.,[[[[.,.],.],.],[.,.]]]=>[2,3,4,6,5,1]=>[1,2,3,4,6,5]=>1
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>[1,5,4,3,2,6]=>1
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>[1,5,4,2,3,6]=>1
[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>[1,5,2,4,3,6]=>1
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>[1,5,3,2,4,6]=>1
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>[1,5,2,3,4,6]=>1
[.,[[[.,.],[.,[.,.]]],.]]=>[2,5,4,3,6,1]=>[1,2,5,4,3,6]=>1
[.,[[[.,.],[[.,.],.]],.]]=>[2,4,5,3,6,1]=>[1,2,5,3,4,6]=>1
[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>[1,3,2,5,4,6]=>1
[.,[[[[.,.],.],[.,.]],.]]=>[2,3,5,4,6,1]=>[1,2,3,5,4,6]=>1
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>[1,4,3,2,5,6]=>1
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>[1,4,2,3,5,6]=>1
[.,[[[[.,.],[.,.]],.],.]]=>[2,4,3,5,6,1]=>[1,2,4,3,5,6]=>1
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>[1,3,2,4,5,6]=>1
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>[1,2,3,4,5,6]=>1
[[.,.],[.,[.,[.,[.,.]]]]]=>[1,6,5,4,3,2]=>[2,1,6,5,4,3]=>2
[[.,.],[.,[.,[[.,.],.]]]]=>[1,5,6,4,3,2]=>[2,1,6,5,3,4]=>2
[[.,.],[.,[[.,.],[.,.]]]]=>[1,4,6,5,3,2]=>[2,1,6,3,5,4]=>2
[[.,.],[.,[[.,[.,.]],.]]]=>[1,5,4,6,3,2]=>[2,1,6,4,3,5]=>2
[[.,.],[.,[[[.,.],.],.]]]=>[1,4,5,6,3,2]=>[2,1,6,3,4,5]=>2
[[.,.],[[.,.],[.,[.,.]]]]=>[1,3,6,5,4,2]=>[2,1,3,6,5,4]=>2
[[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2]=>[2,1,3,6,4,5]=>2
[[.,.],[[.,[.,.]],[.,.]]]=>[1,4,3,6,5,2]=>[2,1,4,3,6,5]=>2
[[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2]=>[2,1,3,4,6,5]=>2
[[.,.],[[.,[.,[.,.]]],.]]=>[1,5,4,3,6,2]=>[2,1,5,4,3,6]=>2
[[.,.],[[.,[[.,.],.]],.]]=>[1,4,5,3,6,2]=>[2,1,5,3,4,6]=>2
[[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2]=>[2,1,3,5,4,6]=>2
[[.,.],[[[.,[.,.]],.],.]]=>[1,4,3,5,6,2]=>[2,1,4,3,5,6]=>2
[[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2]=>[2,1,3,4,5,6]=>2
[[.,[.,.]],[.,[.,[.,.]]]]=>[2,1,6,5,4,3]=>[3,2,1,6,5,4]=>3
[[.,[.,.]],[.,[[.,.],.]]]=>[2,1,5,6,4,3]=>[3,2,1,6,4,5]=>3
[[.,[.,.]],[[.,.],[.,.]]]=>[2,1,4,6,5,3]=>[3,2,1,4,6,5]=>3
[[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3]=>[3,2,1,5,4,6]=>3
[[.,[.,.]],[[[.,.],.],.]]=>[2,1,4,5,6,3]=>[3,2,1,4,5,6]=>3
[[[.,.],.],[.,[.,[.,.]]]]=>[1,2,6,5,4,3]=>[2,3,1,6,5,4]=>3
[[[.,.],.],[.,[[.,.],.]]]=>[1,2,5,6,4,3]=>[2,3,1,6,4,5]=>3
[[[.,.],.],[[.,.],[.,.]]]=>[1,2,4,6,5,3]=>[2,3,1,4,6,5]=>3
[[[.,.],.],[[.,[.,.]],.]]=>[1,2,5,4,6,3]=>[2,3,1,5,4,6]=>3
[[[.,.],.],[[[.,.],.],.]]=>[1,2,4,5,6,3]=>[2,3,1,4,5,6]=>3
[[.,[.,[.,.]]],[.,[.,.]]]=>[3,2,1,6,5,4]=>[4,3,2,1,6,5]=>4
[[.,[.,[.,.]]],[[.,.],.]]=>[3,2,1,5,6,4]=>[4,3,2,1,5,6]=>4
[[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4]=>[4,2,3,1,6,5]=>4
[[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4]=>[4,2,3,1,5,6]=>4
[[[.,.],[.,.]],[.,[.,.]]]=>[1,3,2,6,5,4]=>[2,4,3,1,6,5]=>4
[[[.,.],[.,.]],[[.,.],.]]=>[1,3,2,5,6,4]=>[2,4,3,1,5,6]=>4
[[[.,[.,.]],.],[.,[.,.]]]=>[2,1,3,6,5,4]=>[3,2,4,1,6,5]=>4
[[[.,[.,.]],.],[[.,.],.]]=>[2,1,3,5,6,4]=>[3,2,4,1,5,6]=>4
[[[[.,.],.],.],[.,[.,.]]]=>[1,2,3,6,5,4]=>[2,3,4,1,6,5]=>4
[[[[.,.],.],.],[[.,.],.]]=>[1,2,3,5,6,4]=>[2,3,4,1,5,6]=>4
[[.,[.,[.,[.,.]]]],[.,.]]=>[4,3,2,1,6,5]=>[5,4,3,2,1,6]=>5
[[.,[.,[[.,.],.]]],[.,.]]=>[3,4,2,1,6,5]=>[5,4,2,3,1,6]=>5
[[.,[[.,.],[.,.]]],[.,.]]=>[2,4,3,1,6,5]=>[5,2,4,3,1,6]=>5
[[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>[5,3,2,4,1,6]=>5
[[.,[[[.,.],.],.]],[.,.]]=>[2,3,4,1,6,5]=>[5,2,3,4,1,6]=>5
[[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5]=>[2,5,4,3,1,6]=>5
[[[.,.],[[.,.],.]],[.,.]]=>[1,3,4,2,6,5]=>[2,5,3,4,1,6]=>5
[[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>[3,2,5,4,1,6]=>5
[[[[.,.],.],[.,.]],[.,.]]=>[1,2,4,3,6,5]=>[2,3,5,4,1,6]=>5
[[[.,[.,[.,.]]],.],[.,.]]=>[3,2,1,4,6,5]=>[4,3,2,5,1,6]=>5
[[[.,[[.,.],.]],.],[.,.]]=>[2,3,1,4,6,5]=>[4,2,3,5,1,6]=>5
[[[[.,.],[.,.]],.],[.,.]]=>[1,3,2,4,6,5]=>[2,4,3,5,1,6]=>5
[[[[.,[.,.]],.],.],[.,.]]=>[2,1,3,4,6,5]=>[3,2,4,5,1,6]=>5
[[[[[.,.],.],.],.],[.,.]]=>[1,2,3,4,6,5]=>[2,3,4,5,1,6]=>5
[[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>[6,5,4,3,2,1]=>6
[[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>[6,5,4,2,3,1]=>6
[[.,[.,[[.,.],[.,.]]]],.]=>[3,5,4,2,1,6]=>[6,5,2,4,3,1]=>6
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>[6,5,3,2,4,1]=>6
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>[6,5,2,3,4,1]=>6
[[.,[[.,.],[.,[.,.]]]],.]=>[2,5,4,3,1,6]=>[6,2,5,4,3,1]=>6
[[.,[[.,.],[[.,.],.]]],.]=>[2,4,5,3,1,6]=>[6,2,5,3,4,1]=>6
[[.,[[.,[.,.]],[.,.]]],.]=>[3,2,5,4,1,6]=>[6,3,2,5,4,1]=>6
[[.,[[[.,.],.],[.,.]]],.]=>[2,3,5,4,1,6]=>[6,2,3,5,4,1]=>6
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>[6,4,3,2,5,1]=>6
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>[6,4,2,3,5,1]=>6
[[.,[[[.,.],[.,.]],.]],.]=>[2,4,3,5,1,6]=>[6,2,4,3,5,1]=>6
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>[6,3,2,4,5,1]=>6
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>[6,2,3,4,5,1]=>6
[[[.,.],[.,[.,[.,.]]]],.]=>[1,5,4,3,2,6]=>[2,6,5,4,3,1]=>6
[[[.,.],[.,[[.,.],.]]],.]=>[1,4,5,3,2,6]=>[2,6,5,3,4,1]=>6
[[[.,.],[[.,.],[.,.]]],.]=>[1,3,5,4,2,6]=>[2,6,3,5,4,1]=>6
[[[.,.],[[.,[.,.]],.]],.]=>[1,4,3,5,2,6]=>[2,6,4,3,5,1]=>6
[[[.,.],[[[.,.],.],.]],.]=>[1,3,4,5,2,6]=>[2,6,3,4,5,1]=>6
[[[.,[.,.]],[.,[.,.]]],.]=>[2,1,5,4,3,6]=>[3,2,6,5,4,1]=>6
[[[.,[.,.]],[[.,.],.]],.]=>[2,1,4,5,3,6]=>[3,2,6,4,5,1]=>6
[[[[.,.],.],[.,[.,.]]],.]=>[1,2,5,4,3,6]=>[2,3,6,5,4,1]=>6
[[[[.,.],.],[[.,.],.]],.]=>[1,2,4,5,3,6]=>[2,3,6,4,5,1]=>6
[[[.,[.,[.,.]]],[.,.]],.]=>[3,2,1,5,4,6]=>[4,3,2,6,5,1]=>6
[[[.,[[.,.],.]],[.,.]],.]=>[2,3,1,5,4,6]=>[4,2,3,6,5,1]=>6
[[[[.,.],[.,.]],[.,.]],.]=>[1,3,2,5,4,6]=>[2,4,3,6,5,1]=>6
[[[[.,[.,.]],.],[.,.]],.]=>[2,1,3,5,4,6]=>[3,2,4,6,5,1]=>6
[[[[[.,.],.],.],[.,.]],.]=>[1,2,3,5,4,6]=>[2,3,4,6,5,1]=>6
[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>[5,4,3,2,6,1]=>6
[[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>[5,4,2,3,6,1]=>6
[[[.,[[.,.],[.,.]]],.],.]=>[2,4,3,1,5,6]=>[5,2,4,3,6,1]=>6
[[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>[5,3,2,4,6,1]=>6
[[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>[5,2,3,4,6,1]=>6
[[[[.,.],[.,[.,.]]],.],.]=>[1,4,3,2,5,6]=>[2,5,4,3,6,1]=>6
[[[[.,.],[[.,.],.]],.],.]=>[1,3,4,2,5,6]=>[2,5,3,4,6,1]=>6
[[[[.,[.,.]],[.,.]],.],.]=>[2,1,4,3,5,6]=>[3,2,5,4,6,1]=>6
[[[[[.,.],.],[.,.]],.],.]=>[1,2,4,3,5,6]=>[2,3,5,4,6,1]=>6
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>[4,3,2,5,6,1]=>6
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>[4,2,3,5,6,1]=>6
[[[[[.,.],[.,.]],.],.],.]=>[1,3,2,4,5,6]=>[2,4,3,5,6,1]=>6
[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>[3,2,4,5,6,1]=>6
[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>[2,3,4,5,6,1]=>6
[.,[[.,.],[.,[.,[.,[.,.]]]]]]=>[2,7,6,5,4,3,1]=>[1,2,7,6,5,4,3]=>1
[.,[[.,.],[.,[.,[[.,.],.]]]]]=>[2,6,7,5,4,3,1]=>[1,2,7,6,5,3,4]=>1
[.,[[.,.],[.,[[.,.],[.,.]]]]]=>[2,5,7,6,4,3,1]=>[1,2,7,6,3,5,4]=>1
[.,[[.,.],[.,[[.,[.,.]],.]]]]=>[2,6,5,7,4,3,1]=>[1,2,7,6,4,3,5]=>1
[.,[[.,.],[.,[[[.,.],.],.]]]]=>[2,5,6,7,4,3,1]=>[1,2,7,6,3,4,5]=>1
[.,[[.,.],[[.,.],[.,[.,.]]]]]=>[2,4,7,6,5,3,1]=>[1,2,7,3,6,5,4]=>1
[.,[[.,.],[[.,.],[[.,.],.]]]]=>[2,4,6,7,5,3,1]=>[1,2,7,3,6,4,5]=>1
[.,[[.,.],[[.,[.,.]],[.,.]]]]=>[2,5,4,7,6,3,1]=>[1,2,7,4,3,6,5]=>1
[.,[[.,.],[[[.,.],.],[.,.]]]]=>[2,4,5,7,6,3,1]=>[1,2,7,3,4,6,5]=>1
[.,[[.,.],[[.,[.,[.,.]]],.]]]=>[2,6,5,4,7,3,1]=>[1,2,7,5,4,3,6]=>1
[.,[[.,.],[[.,[[.,.],.]],.]]]=>[2,5,6,4,7,3,1]=>[1,2,7,5,3,4,6]=>1
[.,[[.,.],[[[.,.],[.,.]],.]]]=>[2,4,6,5,7,3,1]=>[1,2,7,3,5,4,6]=>1
[.,[[.,.],[[[.,[.,.]],.],.]]]=>[2,5,4,6,7,3,1]=>[1,2,7,4,3,5,6]=>1
[.,[[.,.],[[[[.,.],.],.],.]]]=>[2,4,5,6,7,3,1]=>[1,2,7,3,4,5,6]=>1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]=>[3,2,7,6,5,4,1]=>[1,3,2,7,6,5,4]=>1
[.,[[.,[.,.]],[.,[[.,.],.]]]]=>[3,2,6,7,5,4,1]=>[1,3,2,7,6,4,5]=>1
[.,[[.,[.,.]],[[.,.],[.,.]]]]=>[3,2,5,7,6,4,1]=>[1,3,2,7,4,6,5]=>1
[.,[[.,[.,.]],[[.,[.,.]],.]]]=>[3,2,6,5,7,4,1]=>[1,3,2,7,5,4,6]=>1
[.,[[.,[.,.]],[[[.,.],.],.]]]=>[3,2,5,6,7,4,1]=>[1,3,2,7,4,5,6]=>1
[.,[[[.,.],.],[.,[.,[.,.]]]]]=>[2,3,7,6,5,4,1]=>[1,2,3,7,6,5,4]=>1
[.,[[[.,.],.],[.,[[.,.],.]]]]=>[2,3,6,7,5,4,1]=>[1,2,3,7,6,4,5]=>1
[.,[[[.,.],.],[[.,.],[.,.]]]]=>[2,3,5,7,6,4,1]=>[1,2,3,7,4,6,5]=>1
[.,[[[.,.],.],[[.,[.,.]],.]]]=>[2,3,6,5,7,4,1]=>[1,2,3,7,5,4,6]=>1
[.,[[[.,.],.],[[[.,.],.],.]]]=>[2,3,5,6,7,4,1]=>[1,2,3,7,4,5,6]=>1
[.,[[.,[.,[.,.]]],[.,[.,.]]]]=>[4,3,2,7,6,5,1]=>[1,4,3,2,7,6,5]=>1
[.,[[.,[.,[.,.]]],[[.,.],.]]]=>[4,3,2,6,7,5,1]=>[1,4,3,2,7,5,6]=>1
[.,[[.,[[.,.],.]],[.,[.,.]]]]=>[3,4,2,7,6,5,1]=>[1,4,2,3,7,6,5]=>1
[.,[[.,[[.,.],.]],[[.,.],.]]]=>[3,4,2,6,7,5,1]=>[1,4,2,3,7,5,6]=>1
[.,[[[.,.],[.,.]],[.,[.,.]]]]=>[2,4,3,7,6,5,1]=>[1,2,4,3,7,6,5]=>1
[.,[[[.,.],[.,.]],[[.,.],.]]]=>[2,4,3,6,7,5,1]=>[1,2,4,3,7,5,6]=>1
[.,[[[.,[.,.]],.],[.,[.,.]]]]=>[3,2,4,7,6,5,1]=>[1,3,2,4,7,6,5]=>1
[.,[[[.,[.,.]],.],[[.,.],.]]]=>[3,2,4,6,7,5,1]=>[1,3,2,4,7,5,6]=>1
[.,[[[[.,.],.],.],[.,[.,.]]]]=>[2,3,4,7,6,5,1]=>[1,2,3,4,7,6,5]=>1
[.,[[[[.,.],.],.],[[.,.],.]]]=>[2,3,4,6,7,5,1]=>[1,2,3,4,7,5,6]=>1
[.,[[[.,.],[.,[.,.]]],[.,.]]]=>[2,5,4,3,7,6,1]=>[1,2,5,4,3,7,6]=>1
[.,[[[.,.],[[.,.],.]],[.,.]]]=>[2,4,5,3,7,6,1]=>[1,2,5,3,4,7,6]=>1
[.,[[[.,[.,.]],[.,.]],[.,.]]]=>[3,2,5,4,7,6,1]=>[1,3,2,5,4,7,6]=>1
[.,[[[[.,.],.],[.,.]],[.,.]]]=>[2,3,5,4,7,6,1]=>[1,2,3,5,4,7,6]=>1
[.,[[[.,[.,[.,.]]],.],[.,.]]]=>[4,3,2,5,7,6,1]=>[1,4,3,2,5,7,6]=>1
[.,[[[.,[[.,.],.]],.],[.,.]]]=>[3,4,2,5,7,6,1]=>[1,4,2,3,5,7,6]=>1
[.,[[[[.,.],[.,.]],.],[.,.]]]=>[2,4,3,5,7,6,1]=>[1,2,4,3,5,7,6]=>1
[.,[[[[.,[.,.]],.],.],[.,.]]]=>[3,2,4,5,7,6,1]=>[1,3,2,4,5,7,6]=>1
[.,[[[[[.,.],.],.],.],[.,.]]]=>[2,3,4,5,7,6,1]=>[1,2,3,4,5,7,6]=>1
[.,[[[.,.],[.,[.,[.,.]]]],.]]=>[2,6,5,4,3,7,1]=>[1,2,6,5,4,3,7]=>1
[.,[[[.,.],[.,[[.,.],.]]],.]]=>[2,5,6,4,3,7,1]=>[1,2,6,5,3,4,7]=>1
[.,[[[.,.],[[.,.],[.,.]]],.]]=>[2,4,6,5,3,7,1]=>[1,2,6,3,5,4,7]=>1
[.,[[[.,.],[[.,[.,.]],.]],.]]=>[2,5,4,6,3,7,1]=>[1,2,6,4,3,5,7]=>1
[.,[[[.,.],[[[.,.],.],.]],.]]=>[2,4,5,6,3,7,1]=>[1,2,6,3,4,5,7]=>1
[.,[[[.,[.,.]],[.,[.,.]]],.]]=>[3,2,6,5,4,7,1]=>[1,3,2,6,5,4,7]=>1
[.,[[[.,[.,.]],[[.,.],.]],.]]=>[3,2,5,6,4,7,1]=>[1,3,2,6,4,5,7]=>1
[.,[[[[.,.],.],[.,[.,.]]],.]]=>[2,3,6,5,4,7,1]=>[1,2,3,6,5,4,7]=>1
[.,[[[[.,.],.],[[.,.],.]],.]]=>[2,3,5,6,4,7,1]=>[1,2,3,6,4,5,7]=>1
[.,[[[.,[.,[.,.]]],[.,.]],.]]=>[4,3,2,6,5,7,1]=>[1,4,3,2,6,5,7]=>1
[.,[[[.,[[.,.],.]],[.,.]],.]]=>[3,4,2,6,5,7,1]=>[1,4,2,3,6,5,7]=>1
[.,[[[[.,.],[.,.]],[.,.]],.]]=>[2,4,3,6,5,7,1]=>[1,2,4,3,6,5,7]=>1
[.,[[[[.,[.,.]],.],[.,.]],.]]=>[3,2,4,6,5,7,1]=>[1,3,2,4,6,5,7]=>1
[.,[[[[[.,.],.],.],[.,.]],.]]=>[2,3,4,6,5,7,1]=>[1,2,3,4,6,5,7]=>1
[.,[[[[.,.],[.,[.,.]]],.],.]]=>[2,5,4,3,6,7,1]=>[1,2,5,4,3,6,7]=>1
[.,[[[[.,.],[[.,.],.]],.],.]]=>[2,4,5,3,6,7,1]=>[1,2,5,3,4,6,7]=>1
[.,[[[[.,[.,.]],[.,.]],.],.]]=>[3,2,5,4,6,7,1]=>[1,3,2,5,4,6,7]=>1
[.,[[[[[.,.],.],[.,.]],.],.]]=>[2,3,5,4,6,7,1]=>[1,2,3,5,4,6,7]=>1
[.,[[[[.,[.,[.,.]]],.],.],.]]=>[4,3,2,5,6,7,1]=>[1,4,3,2,5,6,7]=>1
[.,[[[[.,[[.,.],.]],.],.],.]]=>[3,4,2,5,6,7,1]=>[1,4,2,3,5,6,7]=>1
[.,[[[[[.,.],[.,.]],.],.],.]]=>[2,4,3,5,6,7,1]=>[1,2,4,3,5,6,7]=>1
[.,[[[[[.,[.,.]],.],.],.],.]]=>[3,2,4,5,6,7,1]=>[1,3,2,4,5,6,7]=>1
[.,[[[[[[.,.],.],.],.],.],.]]=>[2,3,4,5,6,7,1]=>[1,2,3,4,5,6,7]=>1
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Description
The size of the first part in the decomposition of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is defined to be the smallest $k > 0$ such that $\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$. This statistic is undefined for the empty permutation.
For the number of parts in the decomposition see St000056The decomposition (or block) number of a permutation..
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is defined to be the smallest $k > 0$ such that $\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$. This statistic is undefined for the empty permutation.
For the number of parts in the decomposition see St000056The decomposition (or block) number of a permutation..
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
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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