Identifier
Values
=>
Cc0010;cc-rep-0
[.,[.,.]]=>[2,1]=>1 [[.,.],.]=>[1,2]=>0 [.,[.,[.,.]]]=>[3,2,1]=>2 [.,[[.,.],.]]=>[2,3,1]=>1 [[.,.],[.,.]]=>[3,1,2]=>1 [[.,[.,.]],.]=>[2,1,3]=>1 [[[.,.],.],.]=>[1,2,3]=>0 [.,[.,[.,[.,.]]]]=>[4,3,2,1]=>3 [.,[.,[[.,.],.]]]=>[3,4,2,1]=>2 [.,[[.,.],[.,.]]]=>[4,2,3,1]=>2 [.,[[.,[.,.]],.]]=>[3,2,4,1]=>2 [.,[[[.,.],.],.]]=>[2,3,4,1]=>1 [[.,.],[.,[.,.]]]=>[4,3,1,2]=>2 [[.,.],[[.,.],.]]=>[3,4,1,2]=>1 [[.,[.,.]],[.,.]]=>[4,2,1,3]=>2 [[[.,.],.],[.,.]]=>[4,1,2,3]=>1 [[.,[.,[.,.]]],.]=>[3,2,1,4]=>2 [[.,[[.,.],.]],.]=>[2,3,1,4]=>1 [[[.,.],[.,.]],.]=>[3,1,2,4]=>1 [[[.,[.,.]],.],.]=>[2,1,3,4]=>1 [[[[.,.],.],.],.]=>[1,2,3,4]=>0 [.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>4 [.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>3 [.,[.,[[.,.],[.,.]]]]=>[5,3,4,2,1]=>3 [.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>3 [.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>2 [.,[[.,.],[.,[.,.]]]]=>[5,4,2,3,1]=>3 [.,[[.,.],[[.,.],.]]]=>[4,5,2,3,1]=>2 [.,[[.,[.,.]],[.,.]]]=>[5,3,2,4,1]=>3 [.,[[[.,.],.],[.,.]]]=>[5,2,3,4,1]=>2 [.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>3 [.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>2 [.,[[[.,.],[.,.]],.]]=>[4,2,3,5,1]=>2 [.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>2 [.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>1 [[.,.],[.,[.,[.,.]]]]=>[5,4,3,1,2]=>3 [[.,.],[.,[[.,.],.]]]=>[4,5,3,1,2]=>2 [[.,.],[[.,.],[.,.]]]=>[5,3,4,1,2]=>2 [[.,.],[[.,[.,.]],.]]=>[4,3,5,1,2]=>2 [[.,.],[[[.,.],.],.]]=>[3,4,5,1,2]=>1 [[.,[.,.]],[.,[.,.]]]=>[5,4,2,1,3]=>3 [[.,[.,.]],[[.,.],.]]=>[4,5,2,1,3]=>2 [[[.,.],.],[.,[.,.]]]=>[5,4,1,2,3]=>2 [[[.,.],.],[[.,.],.]]=>[4,5,1,2,3]=>1 [[.,[.,[.,.]]],[.,.]]=>[5,3,2,1,4]=>3 [[.,[[.,.],.]],[.,.]]=>[5,2,3,1,4]=>2 [[[.,.],[.,.]],[.,.]]=>[5,3,1,2,4]=>2 [[[.,[.,.]],.],[.,.]]=>[5,2,1,3,4]=>2 [[[[.,.],.],.],[.,.]]=>[5,1,2,3,4]=>1 [[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>3 [[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>2 [[.,[[.,.],[.,.]]],.]=>[4,2,3,1,5]=>2 [[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>2 [[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>1 [[[.,.],[.,[.,.]]],.]=>[4,3,1,2,5]=>2 [[[.,.],[[.,.],.]],.]=>[3,4,1,2,5]=>1 [[[.,[.,.]],[.,.]],.]=>[4,2,1,3,5]=>2 [[[[.,.],.],[.,.]],.]=>[4,1,2,3,5]=>1 [[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>2 [[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>1 [[[[.,.],[.,.]],.],.]=>[3,1,2,4,5]=>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 [.,[.,[.,[[.,.],[.,.]]]]]=>[6,4,5,3,2,1]=>4 [.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>4 [.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>3 [.,[.,[[.,.],[.,[.,.]]]]]=>[6,5,3,4,2,1]=>4 [.,[.,[[.,.],[[.,.],.]]]]=>[5,6,3,4,2,1]=>3 [.,[.,[[.,[.,.]],[.,.]]]]=>[6,4,3,5,2,1]=>4 [.,[.,[[[.,.],.],[.,.]]]]=>[6,3,4,5,2,1]=>3 [.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>4 [.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>3 [.,[.,[[[.,.],[.,.]],.]]]=>[5,3,4,6,2,1]=>3 [.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>3 [.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>2 [.,[[.,.],[.,[.,[.,.]]]]]=>[6,5,4,2,3,1]=>4 [.,[[.,.],[.,[[.,.],.]]]]=>[5,6,4,2,3,1]=>3 [.,[[.,.],[[.,.],[.,.]]]]=>[6,4,5,2,3,1]=>3 [.,[[.,.],[[.,[.,.]],.]]]=>[5,4,6,2,3,1]=>3 [.,[[.,.],[[[.,.],.],.]]]=>[4,5,6,2,3,1]=>2 [.,[[.,[.,.]],[.,[.,.]]]]=>[6,5,3,2,4,1]=>4 [.,[[.,[.,.]],[[.,.],.]]]=>[5,6,3,2,4,1]=>3 [.,[[[.,.],.],[.,[.,.]]]]=>[6,5,2,3,4,1]=>3 [.,[[[.,.],.],[[.,.],.]]]=>[5,6,2,3,4,1]=>2 [.,[[.,[.,[.,.]]],[.,.]]]=>[6,4,3,2,5,1]=>4 [.,[[.,[[.,.],.]],[.,.]]]=>[6,3,4,2,5,1]=>3 [.,[[[.,.],[.,.]],[.,.]]]=>[6,4,2,3,5,1]=>3 [.,[[[.,[.,.]],.],[.,.]]]=>[6,3,2,4,5,1]=>3 [.,[[[[.,.],.],.],[.,.]]]=>[6,2,3,4,5,1]=>2 [.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>4 [.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>3 [.,[[.,[[.,.],[.,.]]],.]]=>[5,3,4,2,6,1]=>3 [.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>3 [.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>2 [.,[[[.,.],[.,[.,.]]],.]]=>[5,4,2,3,6,1]=>3 [.,[[[.,.],[[.,.],.]],.]]=>[4,5,2,3,6,1]=>2 [.,[[[.,[.,.]],[.,.]],.]]=>[5,3,2,4,6,1]=>3 [.,[[[[.,.],.],[.,.]],.]]=>[5,2,3,4,6,1]=>2 [.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>3 [.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>2 [.,[[[[.,.],[.,.]],.],.]]=>[4,2,3,5,6,1]=>2 [.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>2 [.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>1 [[.,.],[.,[.,[.,[.,.]]]]]=>[6,5,4,3,1,2]=>4 [[.,.],[.,[.,[[.,.],.]]]]=>[5,6,4,3,1,2]=>3 [[.,.],[.,[[.,.],[.,.]]]]=>[6,4,5,3,1,2]=>3 [[.,.],[.,[[.,[.,.]],.]]]=>[5,4,6,3,1,2]=>3 [[.,.],[.,[[[.,.],.],.]]]=>[4,5,6,3,1,2]=>2 [[.,.],[[.,.],[.,[.,.]]]]=>[6,5,3,4,1,2]=>3 [[.,.],[[.,.],[[.,.],.]]]=>[5,6,3,4,1,2]=>2 [[.,.],[[.,[.,.]],[.,.]]]=>[6,4,3,5,1,2]=>3 [[.,.],[[[.,.],.],[.,.]]]=>[6,3,4,5,1,2]=>2 [[.,.],[[.,[.,[.,.]]],.]]=>[5,4,3,6,1,2]=>3 [[.,.],[[.,[[.,.],.]],.]]=>[4,5,3,6,1,2]=>2 [[.,.],[[[.,.],[.,.]],.]]=>[5,3,4,6,1,2]=>2 [[.,.],[[[.,[.,.]],.],.]]=>[4,3,5,6,1,2]=>2 [[.,.],[[[[.,.],.],.],.]]=>[3,4,5,6,1,2]=>1 [[.,[.,.]],[.,[.,[.,.]]]]=>[6,5,4,2,1,3]=>4 [[.,[.,.]],[.,[[.,.],.]]]=>[5,6,4,2,1,3]=>3 [[.,[.,.]],[[.,.],[.,.]]]=>[6,4,5,2,1,3]=>3 [[.,[.,.]],[[.,[.,.]],.]]=>[5,4,6,2,1,3]=>3 [[.,[.,.]],[[[.,.],.],.]]=>[4,5,6,2,1,3]=>2 [[[.,.],.],[.,[.,[.,.]]]]=>[6,5,4,1,2,3]=>3 [[[.,.],.],[.,[[.,.],.]]]=>[5,6,4,1,2,3]=>2 [[[.,.],.],[[.,.],[.,.]]]=>[6,4,5,1,2,3]=>2 [[[.,.],.],[[.,[.,.]],.]]=>[5,4,6,1,2,3]=>2 [[[.,.],.],[[[.,.],.],.]]=>[4,5,6,1,2,3]=>1 [[.,[.,[.,.]]],[.,[.,.]]]=>[6,5,3,2,1,4]=>4 [[.,[.,[.,.]]],[[.,.],.]]=>[5,6,3,2,1,4]=>3 [[.,[[.,.],.]],[.,[.,.]]]=>[6,5,2,3,1,4]=>3 [[.,[[.,.],.]],[[.,.],.]]=>[5,6,2,3,1,4]=>2 [[[.,.],[.,.]],[.,[.,.]]]=>[6,5,3,1,2,4]=>3 [[[.,.],[.,.]],[[.,.],.]]=>[5,6,3,1,2,4]=>2 [[[.,[.,.]],.],[.,[.,.]]]=>[6,5,2,1,3,4]=>3 [[[.,[.,.]],.],[[.,.],.]]=>[5,6,2,1,3,4]=>2 [[[[.,.],.],.],[.,[.,.]]]=>[6,5,1,2,3,4]=>2 [[[[.,.],.],.],[[.,.],.]]=>[5,6,1,2,3,4]=>1 [[.,[.,[.,[.,.]]]],[.,.]]=>[6,4,3,2,1,5]=>4 [[.,[.,[[.,.],.]]],[.,.]]=>[6,3,4,2,1,5]=>3 [[.,[[.,.],[.,.]]],[.,.]]=>[6,4,2,3,1,5]=>3 [[.,[[.,[.,.]],.]],[.,.]]=>[6,3,2,4,1,5]=>3 [[.,[[[.,.],.],.]],[.,.]]=>[6,2,3,4,1,5]=>2 [[[.,.],[.,[.,.]]],[.,.]]=>[6,4,3,1,2,5]=>3 [[[.,.],[[.,.],.]],[.,.]]=>[6,3,4,1,2,5]=>2 [[[.,[.,.]],[.,.]],[.,.]]=>[6,4,2,1,3,5]=>3 [[[[.,.],.],[.,.]],[.,.]]=>[6,4,1,2,3,5]=>2 [[[.,[.,[.,.]]],.],[.,.]]=>[6,3,2,1,4,5]=>3 [[[.,[[.,.],.]],.],[.,.]]=>[6,2,3,1,4,5]=>2 [[[[.,.],[.,.]],.],[.,.]]=>[6,3,1,2,4,5]=>2 [[[[.,[.,.]],.],.],[.,.]]=>[6,2,1,3,4,5]=>2 [[[[[.,.],.],.],.],[.,.]]=>[6,1,2,3,4,5]=>1 [[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>4 [[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>3 [[.,[.,[[.,.],[.,.]]]],.]=>[5,3,4,2,1,6]=>3 [[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>3 [[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>2 [[.,[[.,.],[.,[.,.]]]],.]=>[5,4,2,3,1,6]=>3 [[.,[[.,.],[[.,.],.]]],.]=>[4,5,2,3,1,6]=>2 [[.,[[.,[.,.]],[.,.]]],.]=>[5,3,2,4,1,6]=>3 [[.,[[[.,.],.],[.,.]]],.]=>[5,2,3,4,1,6]=>2 [[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>3 [[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>2 [[.,[[[.,.],[.,.]],.]],.]=>[4,2,3,5,1,6]=>2 [[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>2 [[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>1 [[[.,.],[.,[.,[.,.]]]],.]=>[5,4,3,1,2,6]=>3 [[[.,.],[.,[[.,.],.]]],.]=>[4,5,3,1,2,6]=>2 [[[.,.],[[.,.],[.,.]]],.]=>[5,3,4,1,2,6]=>2 [[[.,.],[[.,[.,.]],.]],.]=>[4,3,5,1,2,6]=>2 [[[.,.],[[[.,.],.],.]],.]=>[3,4,5,1,2,6]=>1 [[[.,[.,.]],[.,[.,.]]],.]=>[5,4,2,1,3,6]=>3 [[[.,[.,.]],[[.,.],.]],.]=>[4,5,2,1,3,6]=>2 [[[[.,.],.],[.,[.,.]]],.]=>[5,4,1,2,3,6]=>2 [[[[.,.],.],[[.,.],.]],.]=>[4,5,1,2,3,6]=>1 [[[.,[.,[.,.]]],[.,.]],.]=>[5,3,2,1,4,6]=>3 [[[.,[[.,.],.]],[.,.]],.]=>[5,2,3,1,4,6]=>2 [[[[.,.],[.,.]],[.,.]],.]=>[5,3,1,2,4,6]=>2 [[[[.,[.,.]],.],[.,.]],.]=>[5,2,1,3,4,6]=>2 [[[[[.,.],.],.],[.,.]],.]=>[5,1,2,3,4,6]=>1 [[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>3 [[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>2 [[[.,[[.,.],[.,.]]],.],.]=>[4,2,3,1,5,6]=>2 [[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>2 [[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>1 [[[[.,.],[.,[.,.]]],.],.]=>[4,3,1,2,5,6]=>2 [[[[.,.],[[.,.],.]],.],.]=>[3,4,1,2,5,6]=>1 [[[[.,[.,.]],[.,.]],.],.]=>[4,2,1,3,5,6]=>2 [[[[[.,.],.],[.,.]],.],.]=>[4,1,2,3,5,6]=>1 [[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>2 [[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>1 [[[[[.,.],[.,.]],.],.],.]=>[3,1,2,4,5,6]=>1 [[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>1 [[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>0 [[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7]=>0
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Description
The number of recoils of a permutation.
A recoil, or inverse descent of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Map
to 132-avoiding permutation
Description
Return a 132-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 maximal element of the Sylvester class.