Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
St000234: Permutations ⟶ ℤ (values match St000056The decomposition (or block) number of a permutation.)
Values
=>
Cc0010;cc-rep-0
[.,.]=>[1]=>0
[.,[.,.]]=>[2,1]=>0
[[.,.],.]=>[1,2]=>1
[.,[.,[.,.]]]=>[3,2,1]=>0
[.,[[.,.],.]]=>[2,3,1]=>0
[[.,.],[.,.]]=>[1,3,2]=>1
[[.,[.,.]],.]=>[2,1,3]=>1
[[[.,.],.],.]=>[1,2,3]=>2
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>0
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>0
[.,[[.,.],[.,.]]]=>[2,4,3,1]=>0
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>0
[.,[[[.,.],.],.]]=>[2,3,4,1]=>0
[[.,.],[.,[.,.]]]=>[1,4,3,2]=>1
[[.,.],[[.,.],.]]=>[1,3,4,2]=>1
[[.,[.,.]],[.,.]]=>[2,1,4,3]=>1
[[[.,.],.],[.,.]]=>[1,2,4,3]=>2
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>1
[[.,[[.,.],.]],.]=>[2,3,1,4]=>1
[[[.,.],[.,.]],.]=>[1,3,2,4]=>2
[[[.,[.,.]],.],.]=>[2,1,3,4]=>2
[[[[.,.],.],.],.]=>[1,2,3,4]=>3
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>0
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>0
[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>0
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>0
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>0
[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>0
[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>0
[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>0
[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>0
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>0
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>0
[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>0
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>0
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>0
[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>1
[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>1
[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>1
[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>1
[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>1
[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>1
[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>1
[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>2
[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>2
[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>1
[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>1
[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>2
[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>2
[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>3
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>1
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>1
[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>1
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>1
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>1
[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>2
[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>2
[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>2
[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>3
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>2
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>2
[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>3
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>3
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>4
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>0
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>0
[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>0
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>0
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>0
[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>0
[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>0
[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>0
[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>0
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>0
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>0
[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>0
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>0
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>0
[.,[[.,.],[.,[.,[.,.]]]]]=>[2,6,5,4,3,1]=>0
[.,[[.,.],[.,[[.,.],.]]]]=>[2,5,6,4,3,1]=>0
[.,[[.,.],[[.,.],[.,.]]]]=>[2,4,6,5,3,1]=>0
[.,[[.,.],[[.,[.,.]],.]]]=>[2,5,4,6,3,1]=>0
[.,[[.,.],[[[.,.],.],.]]]=>[2,4,5,6,3,1]=>0
[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>0
[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>0
[.,[[[.,.],.],[.,[.,.]]]]=>[2,3,6,5,4,1]=>0
[.,[[[.,.],.],[[.,.],.]]]=>[2,3,5,6,4,1]=>0
[.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>0
[.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>0
[.,[[[.,.],[.,.]],[.,.]]]=>[2,4,3,6,5,1]=>0
[.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>0
[.,[[[[.,.],.],.],[.,.]]]=>[2,3,4,6,5,1]=>0
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>0
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>0
[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>0
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>0
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>0
[.,[[[.,.],[.,[.,.]]],.]]=>[2,5,4,3,6,1]=>0
[.,[[[.,.],[[.,.],.]],.]]=>[2,4,5,3,6,1]=>0
[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>0
[.,[[[[.,.],.],[.,.]],.]]=>[2,3,5,4,6,1]=>0
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>0
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>0
[.,[[[[.,.],[.,.]],.],.]]=>[2,4,3,5,6,1]=>0
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>0
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>0
[[.,.],[.,[.,[.,[.,.]]]]]=>[1,6,5,4,3,2]=>1
[[.,.],[.,[.,[[.,.],.]]]]=>[1,5,6,4,3,2]=>1
[[.,.],[.,[[.,.],[.,.]]]]=>[1,4,6,5,3,2]=>1
[[.,.],[.,[[.,[.,.]],.]]]=>[1,5,4,6,3,2]=>1
[[.,.],[.,[[[.,.],.],.]]]=>[1,4,5,6,3,2]=>1
[[.,.],[[.,.],[.,[.,.]]]]=>[1,3,6,5,4,2]=>1
[[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2]=>1
[[.,.],[[.,[.,.]],[.,.]]]=>[1,4,3,6,5,2]=>1
[[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2]=>1
[[.,.],[[.,[.,[.,.]]],.]]=>[1,5,4,3,6,2]=>1
[[.,.],[[.,[[.,.],.]],.]]=>[1,4,5,3,6,2]=>1
[[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2]=>1
[[.,.],[[[.,[.,.]],.],.]]=>[1,4,3,5,6,2]=>1
[[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2]=>1
[[.,[.,.]],[.,[.,[.,.]]]]=>[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]=>2
[[[.,.],.],[.,[[.,.],.]]]=>[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]=>2
[[.,[.,[.,.]]],[.,[.,.]]]=>[3,2,1,6,5,4]=>1
[[.,[.,[.,.]]],[[.,.],.]]=>[3,2,1,5,6,4]=>1
[[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4]=>1
[[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4]=>1
[[[.,.],[.,.]],[.,[.,.]]]=>[1,3,2,6,5,4]=>2
[[[.,.],[.,.]],[[.,.],.]]=>[1,3,2,5,6,4]=>2
[[[.,[.,.]],.],[.,[.,.]]]=>[2,1,3,6,5,4]=>2
[[[.,[.,.]],.],[[.,.],.]]=>[2,1,3,5,6,4]=>2
[[[[.,.],.],.],[.,[.,.]]]=>[1,2,3,6,5,4]=>3
[[[[.,.],.],.],[[.,.],.]]=>[1,2,3,5,6,4]=>3
[[.,[.,[.,[.,.]]]],[.,.]]=>[4,3,2,1,6,5]=>1
[[.,[.,[[.,.],.]]],[.,.]]=>[3,4,2,1,6,5]=>1
[[.,[[.,.],[.,.]]],[.,.]]=>[2,4,3,1,6,5]=>1
[[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>1
[[.,[[[.,.],.],.]],[.,.]]=>[2,3,4,1,6,5]=>1
[[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5]=>2
[[[.,.],[[.,.],.]],[.,.]]=>[1,3,4,2,6,5]=>2
[[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>2
[[[[.,.],.],[.,.]],[.,.]]=>[1,2,4,3,6,5]=>3
[[[.,[.,[.,.]]],.],[.,.]]=>[3,2,1,4,6,5]=>2
[[[.,[[.,.],.]],.],[.,.]]=>[2,3,1,4,6,5]=>2
[[[[.,.],[.,.]],.],[.,.]]=>[1,3,2,4,6,5]=>3
[[[[.,[.,.]],.],.],[.,.]]=>[2,1,3,4,6,5]=>3
[[[[[.,.],.],.],.],[.,.]]=>[1,2,3,4,6,5]=>4
[[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>1
[[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>1
[[.,[.,[[.,.],[.,.]]]],.]=>[3,5,4,2,1,6]=>1
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>1
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>1
[[.,[[.,.],[.,[.,.]]]],.]=>[2,5,4,3,1,6]=>1
[[.,[[.,.],[[.,.],.]]],.]=>[2,4,5,3,1,6]=>1
[[.,[[.,[.,.]],[.,.]]],.]=>[3,2,5,4,1,6]=>1
[[.,[[[.,.],.],[.,.]]],.]=>[2,3,5,4,1,6]=>1
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>1
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>1
[[.,[[[.,.],[.,.]],.]],.]=>[2,4,3,5,1,6]=>1
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>1
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>1
[[[.,.],[.,[.,[.,.]]]],.]=>[1,5,4,3,2,6]=>2
[[[.,.],[.,[[.,.],.]]],.]=>[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]=>2
[[[.,[.,.]],[.,[.,.]]],.]=>[2,1,5,4,3,6]=>2
[[[.,[.,.]],[[.,.],.]],.]=>[2,1,4,5,3,6]=>2
[[[[.,.],.],[.,[.,.]]],.]=>[1,2,5,4,3,6]=>3
[[[[.,.],.],[[.,.],.]],.]=>[1,2,4,5,3,6]=>3
[[[.,[.,[.,.]]],[.,.]],.]=>[3,2,1,5,4,6]=>2
[[[.,[[.,.],.]],[.,.]],.]=>[2,3,1,5,4,6]=>2
[[[[.,.],[.,.]],[.,.]],.]=>[1,3,2,5,4,6]=>3
[[[[.,[.,.]],.],[.,.]],.]=>[2,1,3,5,4,6]=>3
[[[[[.,.],.],.],[.,.]],.]=>[1,2,3,5,4,6]=>4
[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>2
[[[.,[.,[[.,.],.]]],.],.]=>[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]=>2
[[[[.,.],[.,[.,.]]],.],.]=>[1,4,3,2,5,6]=>3
[[[[.,.],[[.,.],.]],.],.]=>[1,3,4,2,5,6]=>3
[[[[.,[.,.]],[.,.]],.],.]=>[2,1,4,3,5,6]=>3
[[[[[.,.],.],[.,.]],.],.]=>[1,2,4,3,5,6]=>4
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>3
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>3
[[[[[.,.],[.,.]],.],.],.]=>[1,3,2,4,5,6]=>4
[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>4
[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>5
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]=>[7,6,5,4,3,2,1]=>0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]=>[6,7,5,4,3,2,1]=>0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]=>[5,7,6,4,3,2,1]=>0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]=>[6,5,7,4,3,2,1]=>0
[.,[.,[.,[.,[[[.,.],.],.]]]]]=>[5,6,7,4,3,2,1]=>0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]=>[4,7,6,5,3,2,1]=>0
[.,[.,[.,[[.,.],[[.,.],.]]]]]=>[4,6,7,5,3,2,1]=>0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]=>[5,4,7,6,3,2,1]=>0
[.,[.,[.,[[[.,.],.],[.,.]]]]]=>[4,5,7,6,3,2,1]=>0
[.,[.,[.,[[.,[.,[.,.]]],.]]]]=>[6,5,4,7,3,2,1]=>0
[.,[.,[.,[[.,[[.,.],.]],.]]]]=>[5,6,4,7,3,2,1]=>0
[.,[.,[.,[[[.,.],[.,.]],.]]]]=>[4,6,5,7,3,2,1]=>0
[.,[.,[.,[[[.,[.,.]],.],.]]]]=>[5,4,6,7,3,2,1]=>0
[.,[.,[.,[[[[.,.],.],.],.]]]]=>[4,5,6,7,3,2,1]=>0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]=>[3,7,6,5,4,2,1]=>0
[.,[.,[[.,.],[.,[[.,.],.]]]]]=>[3,6,7,5,4,2,1]=>0
[.,[.,[[.,.],[[.,.],[.,.]]]]]=>[3,5,7,6,4,2,1]=>0
[.,[.,[[.,.],[[.,[.,.]],.]]]]=>[3,6,5,7,4,2,1]=>0
[.,[.,[[.,.],[[[.,.],.],.]]]]=>[3,5,6,7,4,2,1]=>0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]=>[4,3,7,6,5,2,1]=>0
[.,[.,[[.,[.,.]],[[.,.],.]]]]=>[4,3,6,7,5,2,1]=>0
[.,[.,[[[.,.],.],[.,[.,.]]]]]=>[3,4,7,6,5,2,1]=>0
[.,[.,[[[.,.],.],[[.,.],.]]]]=>[3,4,6,7,5,2,1]=>0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]=>[5,4,3,7,6,2,1]=>0
[.,[.,[[.,[[.,.],.]],[.,.]]]]=>[4,5,3,7,6,2,1]=>0
[.,[.,[[[.,.],[.,.]],[.,.]]]]=>[3,5,4,7,6,2,1]=>0
[.,[.,[[[.,[.,.]],.],[.,.]]]]=>[4,3,5,7,6,2,1]=>0
[.,[.,[[[[.,.],.],.],[.,.]]]]=>[3,4,5,7,6,2,1]=>0
[.,[.,[[.,[.,[.,[.,.]]]],.]]]=>[6,5,4,3,7,2,1]=>0
[.,[.,[[.,[.,[[.,.],.]]],.]]]=>[5,6,4,3,7,2,1]=>0
[.,[.,[[.,[[.,.],[.,.]]],.]]]=>[4,6,5,3,7,2,1]=>0
[.,[.,[[.,[[.,[.,.]],.]],.]]]=>[5,4,6,3,7,2,1]=>0
[.,[.,[[.,[[[.,.],.],.]],.]]]=>[4,5,6,3,7,2,1]=>0
[.,[.,[[[.,.],[.,[.,.]]],.]]]=>[3,6,5,4,7,2,1]=>0
[.,[.,[[[.,.],[[.,.],.]],.]]]=>[3,5,6,4,7,2,1]=>0
[.,[.,[[[.,[.,.]],[.,.]],.]]]=>[4,3,6,5,7,2,1]=>0
[.,[.,[[[[.,.],.],[.,.]],.]]]=>[3,4,6,5,7,2,1]=>0
[.,[.,[[[.,[.,[.,.]]],.],.]]]=>[5,4,3,6,7,2,1]=>0
[.,[.,[[[.,[[.,.],.]],.],.]]]=>[4,5,3,6,7,2,1]=>0
[.,[.,[[[[.,.],[.,.]],.],.]]]=>[3,5,4,6,7,2,1]=>0
[.,[.,[[[[.,[.,.]],.],.],.]]]=>[4,3,5,6,7,2,1]=>0
[.,[.,[[[[[.,.],.],.],.],.]]]=>[3,4,5,6,7,2,1]=>0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]=>[2,7,6,5,4,3,1]=>0
[.,[[.,.],[.,[.,[[.,.],.]]]]]=>[2,6,7,5,4,3,1]=>0
[.,[[.,.],[.,[[.,.],[.,.]]]]]=>[2,5,7,6,4,3,1]=>0
[.,[[.,.],[.,[[.,[.,.]],.]]]]=>[2,6,5,7,4,3,1]=>0
[.,[[.,.],[.,[[[.,.],.],.]]]]=>[2,5,6,7,4,3,1]=>0
[.,[[.,.],[[.,.],[.,[.,.]]]]]=>[2,4,7,6,5,3,1]=>0
[.,[[.,.],[[.,.],[[.,.],.]]]]=>[2,4,6,7,5,3,1]=>0
[.,[[.,.],[[.,[.,.]],[.,.]]]]=>[2,5,4,7,6,3,1]=>0
[.,[[.,.],[[[.,.],.],[.,.]]]]=>[2,4,5,7,6,3,1]=>0
[.,[[.,.],[[.,[.,[.,.]]],.]]]=>[2,6,5,4,7,3,1]=>0
[.,[[.,.],[[.,[[.,.],.]],.]]]=>[2,5,6,4,7,3,1]=>0
[.,[[.,.],[[[.,.],[.,.]],.]]]=>[2,4,6,5,7,3,1]=>0
[.,[[.,.],[[[.,[.,.]],.],.]]]=>[2,5,4,6,7,3,1]=>0
[.,[[.,.],[[[[.,.],.],.],.]]]=>[2,4,5,6,7,3,1]=>0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]=>[3,2,7,6,5,4,1]=>0
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[[.,.],[.,[.,[.,[[.,.],.]]]]]=>[1,6,7,5,4,3,2]=>1
[[.,.],[.,[.,[[.,.],[.,.]]]]]=>[1,5,7,6,4,3,2]=>1
[[.,.],[.,[.,[[.,[.,.]],.]]]]=>[1,6,5,7,4,3,2]=>1
[[.,.],[.,[.,[[[.,.],.],.]]]]=>[1,5,6,7,4,3,2]=>1
[[.,.],[.,[[.,.],[.,[.,.]]]]]=>[1,4,7,6,5,3,2]=>1
[[.,.],[.,[[.,.],[[.,.],.]]]]=>[1,4,6,7,5,3,2]=>1
[[.,.],[.,[[.,[.,.]],[.,.]]]]=>[1,5,4,7,6,3,2]=>1
[[.,.],[.,[[[.,.],.],[.,.]]]]=>[1,4,5,7,6,3,2]=>1
[[.,.],[.,[[.,[.,[.,.]]],.]]]=>[1,6,5,4,7,3,2]=>1
[[.,.],[.,[[.,[[.,.],.]],.]]]=>[1,5,6,4,7,3,2]=>1
[[.,.],[.,[[[[.,.],.],.],.]]]=>[1,4,5,6,7,3,2]=>1
[[.,.],[[.,.],[.,[.,[.,.]]]]]=>[1,3,7,6,5,4,2]=>1
[[.,.],[[.,.],[.,[[.,.],.]]]]=>[1,3,6,7,5,4,2]=>1
[[.,.],[[.,.],[[.,.],[.,.]]]]=>[1,3,5,7,6,4,2]=>1
[[.,.],[[.,.],[[[.,.],.],.]]]=>[1,3,5,6,7,4,2]=>1
[[.,.],[[[.,.],.],[.,[.,.]]]]=>[1,3,4,7,6,5,2]=>1
[[.,.],[[[.,.],.],[[.,.],.]]]=>[1,3,4,6,7,5,2]=>1
[[.,.],[[.,[.,[.,.]]],[.,.]]]=>[1,5,4,3,7,6,2]=>1
[[.,.],[[.,[[.,.],.]],[.,.]]]=>[1,4,5,3,7,6,2]=>1
[[.,.],[[[[.,.],.],.],[.,.]]]=>[1,3,4,5,7,6,2]=>1
[[.,.],[[.,[.,[.,[.,.]]]],.]]=>[1,6,5,4,3,7,2]=>1
[[.,.],[[.,[.,[[.,.],.]]],.]]=>[1,5,6,4,3,7,2]=>1
[[.,.],[[.,[[.,.],[.,.]]],.]]=>[1,4,6,5,3,7,2]=>1
[[.,.],[[.,[[[.,.],.],.]],.]]=>[1,4,5,6,3,7,2]=>1
[[.,.],[[[.,[.,[.,.]]],.],.]]=>[1,5,4,3,6,7,2]=>1
[[.,.],[[[.,[[.,.],.]],.],.]]=>[1,4,5,3,6,7,2]=>1
[[.,.],[[[[.,[.,.]],.],.],.]]=>[1,4,3,5,6,7,2]=>1
[[.,.],[[[[[.,.],.],.],.],.]]=>[1,3,4,5,6,7,2]=>1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]=>[2,1,7,6,5,4,3]=>1
[[.,[.,.]],[[.,.],[.,[.,.]]]]=>[2,1,4,7,6,5,3]=>1
[[.,[.,.]],[[[[.,.],.],.],.]]=>[2,1,4,5,6,7,3]=>1
[[[.,.],.],[.,[.,[.,[.,.]]]]]=>[1,2,7,6,5,4,3]=>2
[[[.,.],.],[.,[.,[[.,.],.]]]]=>[1,2,6,7,5,4,3]=>2
[[[.,.],.],[.,[[.,.],[.,.]]]]=>[1,2,5,7,6,4,3]=>2
[[[.,.],.],[.,[[[.,.],.],.]]]=>[1,2,5,6,7,4,3]=>2
[[[.,.],.],[[.,.],[.,[.,.]]]]=>[1,2,4,7,6,5,3]=>2
[[[.,.],.],[[.,.],[[.,.],.]]]=>[1,2,4,6,7,5,3]=>2
[[[.,.],.],[[[.,.],.],[.,.]]]=>[1,2,4,5,7,6,3]=>2
[[[.,.],.],[[.,[.,[.,.]]],.]]=>[1,2,6,5,4,7,3]=>2
[[[.,.],.],[[.,[[.,.],.]],.]]=>[1,2,5,6,4,7,3]=>2
[[[.,.],.],[[[.,[.,.]],.],.]]=>[1,2,5,4,6,7,3]=>2
[[[.,.],.],[[[[.,.],.],.],.]]=>[1,2,4,5,6,7,3]=>2
[[.,[.,[.,.]]],[.,[.,[.,.]]]]=>[3,2,1,7,6,5,4]=>1
[[.,[[.,.],.]],[[[.,.],.],.]]=>[2,3,1,5,6,7,4]=>1
[[[.,.],[.,.]],[.,[.,[.,.]]]]=>[1,3,2,7,6,5,4]=>2
[[[.,.],[.,.]],[[[.,.],.],.]]=>[1,3,2,5,6,7,4]=>2
[[[.,[.,.]],.],[.,[.,[.,.]]]]=>[2,1,3,7,6,5,4]=>2
[[[.,[.,.]],.],[[.,.],[.,.]]]=>[2,1,3,5,7,6,4]=>2
[[[.,[.,.]],.],[[[.,.],.],.]]=>[2,1,3,5,6,7,4]=>2
[[[[.,.],.],.],[.,[.,[.,.]]]]=>[1,2,3,7,6,5,4]=>3
[[[[.,.],.],.],[.,[[.,.],.]]]=>[1,2,3,6,7,5,4]=>3
[[[[.,.],.],.],[[.,.],[.,.]]]=>[1,2,3,5,7,6,4]=>3
[[[[.,.],.],.],[[.,[.,.]],.]]=>[1,2,3,6,5,7,4]=>3
[[[[.,.],.],.],[[[.,.],.],.]]=>[1,2,3,5,6,7,4]=>3
[[.,[.,[.,[.,.]]]],[.,[.,.]]]=>[4,3,2,1,7,6,5]=>1
[[.,[[.,.],[.,.]]],[.,[.,.]]]=>[2,4,3,1,7,6,5]=>1
[[.,[[[.,.],.],.]],[.,[.,.]]]=>[2,3,4,1,7,6,5]=>1
[[.,[[[.,.],.],.]],[[.,.],.]]=>[2,3,4,1,6,7,5]=>1
[[[.,.],[.,[.,.]]],[.,[.,.]]]=>[1,4,3,2,7,6,5]=>2
[[[.,.],[[.,.],.]],[.,[.,.]]]=>[1,3,4,2,7,6,5]=>2
[[[.,.],[[.,.],.]],[[.,.],.]]=>[1,3,4,2,6,7,5]=>2
[[[.,[.,.]],[.,.]],[.,[.,.]]]=>[2,1,4,3,7,6,5]=>2
[[[.,[.,.]],[.,.]],[[.,.],.]]=>[2,1,4,3,6,7,5]=>2
[[[[.,.],.],[.,.]],[.,[.,.]]]=>[1,2,4,3,7,6,5]=>3
[[[[.,.],.],[.,.]],[[.,.],.]]=>[1,2,4,3,6,7,5]=>3
[[[.,[.,[.,.]]],.],[.,[.,.]]]=>[3,2,1,4,7,6,5]=>2
[[[.,[[.,.],.]],.],[.,[.,.]]]=>[2,3,1,4,7,6,5]=>2
[[[.,[[.,.],.]],.],[[.,.],.]]=>[2,3,1,4,6,7,5]=>2
[[[[.,.],[.,.]],.],[.,[.,.]]]=>[1,3,2,4,7,6,5]=>3
[[[[.,.],[.,.]],.],[[.,.],.]]=>[1,3,2,4,6,7,5]=>3
[[[[.,[.,.]],.],.],[.,[.,.]]]=>[2,1,3,4,7,6,5]=>3
[[[[.,[.,.]],.],.],[[.,.],.]]=>[2,1,3,4,6,7,5]=>3
[[[[[.,.],.],.],.],[.,[.,.]]]=>[1,2,3,4,7,6,5]=>4
[[[[[.,.],.],.],.],[[.,.],.]]=>[1,2,3,4,6,7,5]=>4
[[.,[.,[.,[.,[.,.]]]]],[.,.]]=>[5,4,3,2,1,7,6]=>1
[[.,[[.,[.,[.,.]]],.]],[.,.]]=>[4,3,2,5,1,7,6]=>1
[[.,[[[[.,.],.],.],.]],[.,.]]=>[2,3,4,5,1,7,6]=>1
[[[.,.],[.,[.,[.,.]]]],[.,.]]=>[1,5,4,3,2,7,6]=>2
[[[.,.],[[[.,.],.],.]],[.,.]]=>[1,3,4,5,2,7,6]=>2
[[[.,[.,.]],[.,[.,.]]],[.,.]]=>[2,1,5,4,3,7,6]=>2
[[[.,[.,.]],[[.,.],.]],[.,.]]=>[2,1,4,5,3,7,6]=>2
[[[[.,.],.],[.,[.,.]]],[.,.]]=>[1,2,5,4,3,7,6]=>3
[[[[.,.],.],[[.,.],.]],[.,.]]=>[1,2,4,5,3,7,6]=>3
[[[.,[.,[.,.]]],[.,.]],[.,.]]=>[3,2,1,5,4,7,6]=>2
[[[.,[[.,.],.]],[.,.]],[.,.]]=>[2,3,1,5,4,7,6]=>2
[[[[.,.],[.,.]],[.,.]],[.,.]]=>[1,3,2,5,4,7,6]=>3
[[[[.,[.,.]],.],[.,.]],[.,.]]=>[2,1,3,5,4,7,6]=>3
[[[[[.,.],.],.],[.,.]],[.,.]]=>[1,2,3,5,4,7,6]=>4
[[[.,[.,[.,[.,.]]]],.],[.,.]]=>[4,3,2,1,5,7,6]=>2
[[[.,[[[.,.],.],.]],.],[.,.]]=>[2,3,4,1,5,7,6]=>2
[[[[.,.],[.,[.,.]]],.],[.,.]]=>[1,4,3,2,5,7,6]=>3
[[[[.,.],[[.,.],.]],.],[.,.]]=>[1,3,4,2,5,7,6]=>3
[[[[.,[.,.]],[.,.]],.],[.,.]]=>[2,1,4,3,5,7,6]=>3
[[[[[.,.],.],[.,.]],.],[.,.]]=>[1,2,4,3,5,7,6]=>4
[[[[.,[.,[.,.]]],.],.],[.,.]]=>[3,2,1,4,5,7,6]=>3
[[[[.,[[.,.],.]],.],.],[.,.]]=>[2,3,1,4,5,7,6]=>3
[[[[[.,.],[.,.]],.],.],[.,.]]=>[1,3,2,4,5,7,6]=>4
[[[[[.,[.,.]],.],.],.],[.,.]]=>[2,1,3,4,5,7,6]=>4
[[[[[[.,.],.],.],.],.],[.,.]]=>[1,2,3,4,5,7,6]=>5
[[.,[.,[.,[.,[.,[.,.]]]]]],.]=>[6,5,4,3,2,1,7]=>1
[[.,[.,[.,[.,[[.,.],.]]]]],.]=>[5,6,4,3,2,1,7]=>1
[[.,[.,[.,[[.,.],[.,.]]]]],.]=>[4,6,5,3,2,1,7]=>1
[[.,[.,[.,[[.,[.,.]],.]]]],.]=>[5,4,6,3,2,1,7]=>1
[[.,[.,[.,[[[.,.],.],.]]]],.]=>[4,5,6,3,2,1,7]=>1
[[.,[.,[[.,.],[.,[.,.]]]]],.]=>[3,6,5,4,2,1,7]=>1
[[.,[.,[[.,.],[[.,.],.]]]],.]=>[3,5,6,4,2,1,7]=>1
[[.,[.,[[.,[.,.]],[.,.]]]],.]=>[4,3,6,5,2,1,7]=>1
[[.,[.,[[.,[.,[.,.]]],.]]],.]=>[5,4,3,6,2,1,7]=>1
[[.,[.,[[.,[[.,.],.]],.]]],.]=>[4,5,3,6,2,1,7]=>1
[[.,[.,[[[.,[.,.]],.],.]]],.]=>[4,3,5,6,2,1,7]=>1
[[.,[.,[[[[.,.],.],.],.]]],.]=>[3,4,5,6,2,1,7]=>1
[[.,[[.,.],[.,[.,[.,.]]]]],.]=>[2,6,5,4,3,1,7]=>1
[[.,[[.,[.,.]],[.,[.,.]]]],.]=>[3,2,6,5,4,1,7]=>1
[[.,[[.,[.,[.,[.,.]]]],.]],.]=>[5,4,3,2,6,1,7]=>1
[[.,[[.,[.,[[.,.],.]]],.]],.]=>[4,5,3,2,6,1,7]=>1
[[.,[[.,[[.,[.,.]],.]],.]],.]=>[4,3,5,2,6,1,7]=>1
[[.,[[.,[[[.,.],.],.]],.]],.]=>[3,4,5,2,6,1,7]=>1
[[.,[[[.,[.,[.,.]]],.],.]],.]=>[4,3,2,5,6,1,7]=>1
[[.,[[[.,[[.,.],.]],.],.]],.]=>[3,4,2,5,6,1,7]=>1
[[.,[[[[.,[.,.]],.],.],.]],.]=>[3,2,4,5,6,1,7]=>1
[[.,[[[[[.,.],.],.],.],.]],.]=>[2,3,4,5,6,1,7]=>1
[[[.,.],[.,[.,[.,[.,.]]]]],.]=>[1,6,5,4,3,2,7]=>2
[[[.,.],[[[[.,.],.],.],.]],.]=>[1,3,4,5,6,2,7]=>2
[[[.,[.,.]],[.,[.,[.,.]]]],.]=>[2,1,6,5,4,3,7]=>2
[[[.,[.,.]],[.,[[.,.],.]]],.]=>[2,1,5,6,4,3,7]=>2
[[[.,[.,.]],[[[.,.],.],.]],.]=>[2,1,4,5,6,3,7]=>2
[[[[.,.],.],[.,[.,[.,.]]]],.]=>[1,2,6,5,4,3,7]=>3
[[[[.,.],.],[[[.,.],.],.]],.]=>[1,2,4,5,6,3,7]=>3
[[[.,[.,[.,.]]],[.,[.,.]]],.]=>[3,2,1,6,5,4,7]=>2
[[[.,[[.,.],.]],[[.,.],.]],.]=>[2,3,1,5,6,4,7]=>2
[[[[.,.],[.,.]],[.,[.,.]]],.]=>[1,3,2,6,5,4,7]=>3
[[[[.,.],[.,.]],[[.,.],.]],.]=>[1,3,2,5,6,4,7]=>3
[[[[.,[.,.]],.],[.,[.,.]]],.]=>[2,1,3,6,5,4,7]=>3
[[[[.,[.,.]],.],[[.,.],.]],.]=>[2,1,3,5,6,4,7]=>3
[[[[[.,.],.],.],[.,[.,.]]],.]=>[1,2,3,6,5,4,7]=>4
[[[[[.,.],.],.],[[.,.],.]],.]=>[1,2,3,5,6,4,7]=>4
[[[.,[.,[.,[.,.]]]],[.,.]],.]=>[4,3,2,1,6,5,7]=>2
[[[.,[[[.,.],.],.]],[.,.]],.]=>[2,3,4,1,6,5,7]=>2
[[[[.,.],[.,[.,.]]],[.,.]],.]=>[1,4,3,2,6,5,7]=>3
[[[[.,.],[[.,.],.]],[.,.]],.]=>[1,3,4,2,6,5,7]=>3
[[[[.,[.,.]],[.,.]],[.,.]],.]=>[2,1,4,3,6,5,7]=>3
[[[[[.,.],.],[.,.]],[.,.]],.]=>[1,2,4,3,6,5,7]=>4
[[[[.,[.,[.,.]]],.],[.,.]],.]=>[3,2,1,4,6,5,7]=>3
[[[[.,[[.,.],.]],.],[.,.]],.]=>[2,3,1,4,6,5,7]=>3
[[[[[.,.],[.,.]],.],[.,.]],.]=>[1,3,2,4,6,5,7]=>4
[[[[[.,[.,.]],.],.],[.,.]],.]=>[2,1,3,4,6,5,7]=>4
[[[[[[.,.],.],.],.],[.,.]],.]=>[1,2,3,4,6,5,7]=>5
[[[.,[.,[.,[.,[.,.]]]]],.],.]=>[5,4,3,2,1,6,7]=>2
[[[.,[.,[.,[[.,.],.]]]],.],.]=>[4,5,3,2,1,6,7]=>2
[[[.,[.,[[.,[.,.]],.]]],.],.]=>[4,3,5,2,1,6,7]=>2
[[[.,[.,[[[.,.],.],.]]],.],.]=>[3,4,5,2,1,6,7]=>2
[[[.,[[.,[.,[.,.]]],.]],.],.]=>[4,3,2,5,1,6,7]=>2
[[[.,[[.,[[.,.],.]],.]],.],.]=>[3,4,2,5,1,6,7]=>2
[[[.,[[[.,[.,.]],.],.]],.],.]=>[3,2,4,5,1,6,7]=>2
[[[.,[[[[.,.],.],.],.]],.],.]=>[2,3,4,5,1,6,7]=>2
[[[[.,.],[.,[.,[.,.]]]],.],.]=>[1,5,4,3,2,6,7]=>3
[[[[.,.],[[[.,.],.],.]],.],.]=>[1,3,4,5,2,6,7]=>3
[[[[.,[.,.]],[.,[.,.]]],.],.]=>[2,1,5,4,3,6,7]=>3
[[[[.,[.,.]],[[.,.],.]],.],.]=>[2,1,4,5,3,6,7]=>3
[[[[[.,.],.],[.,[.,.]]],.],.]=>[1,2,5,4,3,6,7]=>4
[[[[[.,.],.],[[.,.],.]],.],.]=>[1,2,4,5,3,6,7]=>4
[[[[.,[.,[.,.]]],[.,.]],.],.]=>[3,2,1,5,4,6,7]=>3
[[[[.,[[.,.],.]],[.,.]],.],.]=>[2,3,1,5,4,6,7]=>3
[[[[[.,.],[.,.]],[.,.]],.],.]=>[1,3,2,5,4,6,7]=>4
[[[[[.,[.,.]],.],[.,.]],.],.]=>[2,1,3,5,4,6,7]=>4
[[[[[[.,.],.],.],[.,.]],.],.]=>[1,2,3,5,4,6,7]=>5
[[[[.,[.,[.,[.,.]]]],.],.],.]=>[4,3,2,1,5,6,7]=>3
[[[[.,[.,[[.,.],.]]],.],.],.]=>[3,4,2,1,5,6,7]=>3
[[[[.,[[.,[.,.]],.]],.],.],.]=>[3,2,4,1,5,6,7]=>3
[[[[.,[[[.,.],.],.]],.],.],.]=>[2,3,4,1,5,6,7]=>3
[[[[[.,.],[.,[.,.]]],.],.],.]=>[1,4,3,2,5,6,7]=>4
[[[[[.,.],[[.,.],.]],.],.],.]=>[1,3,4,2,5,6,7]=>4
[[[[[.,[.,.]],[.,.]],.],.],.]=>[2,1,4,3,5,6,7]=>4
[[[[[[.,.],.],[.,.]],.],.],.]=>[1,2,4,3,5,6,7]=>5
[[[[[.,[.,[.,.]]],.],.],.],.]=>[3,2,1,4,5,6,7]=>4
[[[[[.,[[.,.],.]],.],.],.],.]=>[2,3,1,4,5,6,7]=>4
[[[[[[.,.],[.,.]],.],.],.],.]=>[1,3,2,4,5,6,7]=>5
[[[[[[.,[.,.]],.],.],.],.],.]=>[2,1,3,4,5,6,7]=>5
[[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7]=>6
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Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives oeis:A003319.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives oeis:A003319.
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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