Identifier
Values
=>
Cc0010;cc-rep-0
[.,.]=>[1]=>0 [.,[.,.]]=>[2,1]=>1 [[.,.],.]=>[1,2]=>0 [.,[.,[.,.]]]=>[3,2,1]=>1 [.,[[.,.],.]]=>[2,3,1]=>2 [[.,.],[.,.]]=>[3,1,2]=>1 [[.,[.,.]],.]=>[2,1,3]=>1 [[[.,.],.],.]=>[1,2,3]=>0 [.,[.,[.,[.,.]]]]=>[4,3,2,1]=>2 [.,[.,[[.,.],.]]]=>[3,4,2,1]=>2 [.,[[.,.],[.,.]]]=>[4,2,3,1]=>1 [.,[[.,[.,.]],.]]=>[3,2,4,1]=>2 [.,[[[.,.],.],.]]=>[2,3,4,1]=>3 [[.,.],[.,[.,.]]]=>[4,3,1,2]=>2 [[.,.],[[.,.],.]]=>[3,4,1,2]=>2 [[.,[.,.]],[.,.]]=>[4,2,1,3]=>1 [[[.,.],.],[.,.]]=>[4,1,2,3]=>1 [[.,[.,[.,.]]],.]=>[3,2,1,4]=>1 [[.,[[.,.],.]],.]=>[2,3,1,4]=>2 [[[.,.],[.,.]],.]=>[3,1,2,4]=>1 [[[.,[.,.]],.],.]=>[2,1,3,4]=>1 [[[[.,.],.],.],.]=>[1,2,3,4]=>0 [.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>2 [.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>2 [.,[.,[[.,.],[.,.]]]]=>[5,3,4,2,1]=>3 [.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>3 [.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>3 [.,[[.,.],[.,[.,.]]]]=>[5,4,2,3,1]=>2 [.,[[.,.],[[.,.],.]]]=>[4,5,2,3,1]=>2 [.,[[.,[.,.]],[.,.]]]=>[5,3,2,4,1]=>2 [.,[[[.,.],.],[.,.]]]=>[5,2,3,4,1]=>1 [.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>3 [.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>3 [.,[[[.,.],[.,.]],.]]=>[4,2,3,5,1]=>2 [.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>3 [.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>4 [[.,.],[.,[.,[.,.]]]]=>[5,4,3,1,2]=>2 [[.,.],[.,[[.,.],.]]]=>[4,5,3,1,2]=>2 [[.,.],[[.,.],[.,.]]]=>[5,3,4,1,2]=>3 [[.,.],[[.,[.,.]],.]]=>[4,3,5,1,2]=>3 [[.,.],[[[.,.],.],.]]=>[3,4,5,1,2]=>3 [[.,[.,.]],[.,[.,.]]]=>[5,4,2,1,3]=>2 [[.,[.,.]],[[.,.],.]]=>[4,5,2,1,3]=>2 [[[.,.],.],[.,[.,.]]]=>[5,4,1,2,3]=>2 [[[.,.],.],[[.,.],.]]=>[4,5,1,2,3]=>2 [[.,[.,[.,.]]],[.,.]]=>[5,3,2,1,4]=>2 [[.,[[.,.],.]],[.,.]]=>[5,2,3,1,4]=>1 [[[.,.],[.,.]],[.,.]]=>[5,3,1,2,4]=>2 [[[.,[.,.]],.],[.,.]]=>[5,2,1,3,4]=>1 [[[[.,.],.],.],[.,.]]=>[5,1,2,3,4]=>1 [[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>2 [[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>2 [[.,[[.,.],[.,.]]],.]=>[4,2,3,1,5]=>1 [[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>2 [[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>3 [[[.,.],[.,[.,.]]],.]=>[4,3,1,2,5]=>2 [[[.,.],[[.,.],.]],.]=>[3,4,1,2,5]=>2 [[[.,[.,.]],[.,.]],.]=>[4,2,1,3,5]=>1 [[[[.,.],.],[.,.]],.]=>[4,1,2,3,5]=>1 [[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>1 [[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>2 [[[[.,.],[.,.]],.],.]=>[3,1,2,4,5]=>1 [[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>1 [[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>0 [.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>3 [.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>3 [.,[.,[.,[[.,.],[.,.]]]]]=>[6,4,5,3,2,1]=>3 [.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>3 [.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>3 [.,[.,[[.,.],[.,[.,.]]]]]=>[6,5,3,4,2,1]=>2 [.,[.,[[.,.],[[.,.],.]]]]=>[5,6,3,4,2,1]=>2 [.,[.,[[.,[.,.]],[.,.]]]]=>[6,4,3,5,2,1]=>3 [.,[.,[[[.,.],.],[.,.]]]]=>[6,3,4,5,2,1]=>4 [.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>3 [.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>3 [.,[.,[[[.,.],[.,.]],.]]]=>[5,3,4,6,2,1]=>4 [.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>4 [.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>4 [.,[[.,.],[.,[.,[.,.]]]]]=>[6,5,4,2,3,1]=>3 [.,[[.,.],[.,[[.,.],.]]]]=>[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]=>3 [.,[[.,[.,.]],[.,[.,.]]]]=>[6,5,3,2,4,1]=>2 [.,[[.,[.,.]],[[.,.],.]]]=>[5,6,3,2,4,1]=>2 [.,[[[.,.],.],[.,[.,.]]]]=>[6,5,2,3,4,1]=>2 [.,[[[.,.],.],[[.,.],.]]]=>[5,6,2,3,4,1]=>2 [.,[[.,[.,[.,.]]],[.,.]]]=>[6,4,3,2,5,1]=>2 [.,[[.,[[.,.],.]],[.,.]]]=>[6,3,4,2,5,1]=>3 [.,[[[.,.],[.,.]],[.,.]]]=>[6,4,2,3,5,1]=>2 [.,[[[.,[.,.]],.],[.,.]]]=>[6,3,2,4,5,1]=>2 [.,[[[[.,.],.],.],[.,.]]]=>[6,2,3,4,5,1]=>1 [.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>3 [.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>3 [.,[[.,[[.,.],[.,.]]],.]]=>[5,3,4,2,6,1]=>4 [.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>4 [.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>4 [.,[[[.,.],[.,[.,.]]],.]]=>[5,4,2,3,6,1]=>3 [.,[[[.,.],[[.,.],.]],.]]=>[4,5,2,3,6,1]=>3 [.,[[[.,[.,.]],[.,.]],.]]=>[5,3,2,4,6,1]=>3 [.,[[[[.,.],.],[.,.]],.]]=>[5,2,3,4,6,1]=>2 [.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>4 [.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>4 [.,[[[[.,.],[.,.]],.],.]]=>[4,2,3,5,6,1]=>3 [.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>4 [.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>5 [[.,.],[.,[.,[.,[.,.]]]]]=>[6,5,4,3,1,2]=>3 [[.,.],[.,[.,[[.,.],.]]]]=>[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]=>3 [[.,.],[[.,.],[.,[.,.]]]]=>[6,5,3,4,1,2]=>2 [[.,.],[[.,.],[[.,.],.]]]=>[5,6,3,4,1,2]=>2 [[.,.],[[.,[.,.]],[.,.]]]=>[6,4,3,5,1,2]=>3 [[.,.],[[[.,.],.],[.,.]]]=>[6,3,4,5,1,2]=>4 [[.,.],[[.,[.,[.,.]]],.]]=>[5,4,3,6,1,2]=>3 [[.,.],[[.,[[.,.],.]],.]]=>[4,5,3,6,1,2]=>3 [[.,.],[[[.,.],[.,.]],.]]=>[5,3,4,6,1,2]=>4 [[.,.],[[[.,[.,.]],.],.]]=>[4,3,5,6,1,2]=>4 [[.,.],[[[[.,.],.],.],.]]=>[3,4,5,6,1,2]=>4 [[.,[.,.]],[.,[.,[.,.]]]]=>[6,5,4,2,1,3]=>3 [[.,[.,.]],[.,[[.,.],.]]]=>[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]=>3 [[[.,.],.],[.,[.,[.,.]]]]=>[6,5,4,1,2,3]=>3 [[[.,.],.],[.,[[.,.],.]]]=>[5,6,4,1,2,3]=>3 [[[.,.],.],[[.,.],[.,.]]]=>[6,4,5,1,2,3]=>3 [[[.,.],.],[[.,[.,.]],.]]=>[5,4,6,1,2,3]=>3 [[[.,.],.],[[[.,.],.],.]]=>[4,5,6,1,2,3]=>3 [[.,[.,[.,.]]],[.,[.,.]]]=>[6,5,3,2,1,4]=>2 [[.,[.,[.,.]]],[[.,.],.]]=>[5,6,3,2,1,4]=>2 [[.,[[.,.],.]],[.,[.,.]]]=>[6,5,2,3,1,4]=>2 [[.,[[.,.],.]],[[.,.],.]]=>[5,6,2,3,1,4]=>2 [[[.,.],[.,.]],[.,[.,.]]]=>[6,5,3,1,2,4]=>2 [[[.,.],[.,.]],[[.,.],.]]=>[5,6,3,1,2,4]=>2 [[[.,[.,.]],.],[.,[.,.]]]=>[6,5,2,1,3,4]=>2 [[[.,[.,.]],.],[[.,.],.]]=>[5,6,2,1,3,4]=>2 [[[[.,.],.],.],[.,[.,.]]]=>[6,5,1,2,3,4]=>2 [[[[.,.],.],.],[[.,.],.]]=>[5,6,1,2,3,4]=>2 [[.,[.,[.,[.,.]]]],[.,.]]=>[6,4,3,2,1,5]=>2 [[.,[.,[[.,.],.]]],[.,.]]=>[6,3,4,2,1,5]=>3 [[.,[[.,.],[.,.]]],[.,.]]=>[6,4,2,3,1,5]=>2 [[.,[[.,[.,.]],.]],[.,.]]=>[6,3,2,4,1,5]=>2 [[.,[[[.,.],.],.]],[.,.]]=>[6,2,3,4,1,5]=>1 [[[.,.],[.,[.,.]]],[.,.]]=>[6,4,3,1,2,5]=>2 [[[.,.],[[.,.],.]],[.,.]]=>[6,3,4,1,2,5]=>3 [[[.,[.,.]],[.,.]],[.,.]]=>[6,4,2,1,3,5]=>2 [[[[.,.],.],[.,.]],[.,.]]=>[6,4,1,2,3,5]=>2 [[[.,[.,[.,.]]],.],[.,.]]=>[6,3,2,1,4,5]=>2 [[[.,[[.,.],.]],.],[.,.]]=>[6,2,3,1,4,5]=>1 [[[[.,.],[.,.]],.],[.,.]]=>[6,3,1,2,4,5]=>2 [[[[.,[.,.]],.],.],[.,.]]=>[6,2,1,3,4,5]=>1 [[[[[.,.],.],.],.],[.,.]]=>[6,1,2,3,4,5]=>1 [[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>2 [[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>2 [[.,[.,[[.,.],[.,.]]]],.]=>[5,3,4,2,1,6]=>3 [[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>3 [[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>3 [[.,[[.,.],[.,[.,.]]]],.]=>[5,4,2,3,1,6]=>2 [[.,[[.,.],[[.,.],.]]],.]=>[4,5,2,3,1,6]=>2 [[.,[[.,[.,.]],[.,.]]],.]=>[5,3,2,4,1,6]=>2 [[.,[[[.,.],.],[.,.]]],.]=>[5,2,3,4,1,6]=>1 [[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>3 [[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>3 [[.,[[[.,.],[.,.]],.]],.]=>[4,2,3,5,1,6]=>2 [[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>3 [[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>4 [[[.,.],[.,[.,[.,.]]]],.]=>[5,4,3,1,2,6]=>2 [[[.,.],[.,[[.,.],.]]],.]=>[4,5,3,1,2,6]=>2 [[[.,.],[[.,.],[.,.]]],.]=>[5,3,4,1,2,6]=>3 [[[.,.],[[.,[.,.]],.]],.]=>[4,3,5,1,2,6]=>3 [[[.,.],[[[.,.],.],.]],.]=>[3,4,5,1,2,6]=>3 [[[.,[.,.]],[.,[.,.]]],.]=>[5,4,2,1,3,6]=>2 [[[.,[.,.]],[[.,.],.]],.]=>[4,5,2,1,3,6]=>2 [[[[.,.],.],[.,[.,.]]],.]=>[5,4,1,2,3,6]=>2 [[[[.,.],.],[[.,.],.]],.]=>[4,5,1,2,3,6]=>2 [[[.,[.,[.,.]]],[.,.]],.]=>[5,3,2,1,4,6]=>2 [[[.,[[.,.],.]],[.,.]],.]=>[5,2,3,1,4,6]=>1 [[[[.,.],[.,.]],[.,.]],.]=>[5,3,1,2,4,6]=>2 [[[[.,[.,.]],.],[.,.]],.]=>[5,2,1,3,4,6]=>1 [[[[[.,.],.],.],[.,.]],.]=>[5,1,2,3,4,6]=>1 [[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>2 [[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>2 [[[.,[[.,.],[.,.]]],.],.]=>[4,2,3,1,5,6]=>1 [[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>2 [[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>3 [[[[.,.],[.,[.,.]]],.],.]=>[4,3,1,2,5,6]=>2 [[[[.,.],[[.,.],.]],.],.]=>[3,4,1,2,5,6]=>2 [[[[.,[.,.]],[.,.]],.],.]=>[4,2,1,3,5,6]=>1 [[[[[.,.],.],[.,.]],.],.]=>[4,1,2,3,5,6]=>1 [[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>1 [[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>2 [[[[[.,.],[.,.]],.],.],.]=>[3,1,2,4,5,6]=>1 [[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>1 [[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>0
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Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $den$ is the Denert index of a permutation, see St000156The Denert index of a permutation..
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.