Identifier
Values
=>
Cc0010;cc-rep-0 Cc0005;cc-rep
[.,.]=>[1,0]=>0 [.,[.,.]]=>[1,1,0,0]=>0 [[.,.],.]=>[1,0,1,0]=>1 [.,[.,[.,.]]]=>[1,1,1,0,0,0]=>0 [.,[[.,.],.]]=>[1,1,0,1,0,0]=>1 [[.,.],[.,.]]=>[1,0,1,1,0,0]=>1 [[.,[.,.]],.]=>[1,1,0,0,1,0]=>2 [[[.,.],.],.]=>[1,0,1,0,1,0]=>3 [.,[.,[.,[.,.]]]]=>[1,1,1,1,0,0,0,0]=>0 [.,[.,[[.,.],.]]]=>[1,1,1,0,1,0,0,0]=>1 [.,[[.,.],[.,.]]]=>[1,1,0,1,1,0,0,0]=>1 [.,[[.,[.,.]],.]]=>[1,1,1,0,0,1,0,0]=>2 [.,[[[.,.],.],.]]=>[1,1,0,1,0,1,0,0]=>2 [[.,.],[.,[.,.]]]=>[1,0,1,1,1,0,0,0]=>1 [[.,.],[[.,.],.]]=>[1,0,1,1,0,1,0,0]=>3 [[.,[.,.]],[.,.]]=>[1,1,0,0,1,1,0,0]=>2 [[[.,.],.],[.,.]]=>[1,0,1,0,1,1,0,0]=>3 [[.,[.,[.,.]]],.]=>[1,1,1,0,0,0,1,0]=>3 [[.,[[.,.],.]],.]=>[1,1,0,1,0,0,1,0]=>4 [[[.,.],[.,.]],.]=>[1,0,1,1,0,0,1,0]=>4 [[[.,[.,.]],.],.]=>[1,1,0,0,1,0,1,0]=>5 [[[[.,.],.],.],.]=>[1,0,1,0,1,0,1,0]=>6 [.,[.,[.,[.,[.,.]]]]]=>[1,1,1,1,1,0,0,0,0,0]=>0 [.,[.,[.,[[.,.],.]]]]=>[1,1,1,1,0,1,0,0,0,0]=>1 [.,[.,[[.,.],[.,.]]]]=>[1,1,1,0,1,1,0,0,0,0]=>1 [.,[.,[[.,[.,.]],.]]]=>[1,1,1,1,0,0,1,0,0,0]=>2 [.,[.,[[[.,.],.],.]]]=>[1,1,1,0,1,0,1,0,0,0]=>2 [.,[[.,.],[.,[.,.]]]]=>[1,1,0,1,1,1,0,0,0,0]=>1 [.,[[.,.],[[.,.],.]]]=>[1,1,0,1,1,0,1,0,0,0]=>2 [.,[[.,[.,.]],[.,.]]]=>[1,1,1,0,0,1,1,0,0,0]=>2 [.,[[[.,.],.],[.,.]]]=>[1,1,0,1,0,1,1,0,0,0]=>2 [.,[[.,[.,[.,.]]],.]]=>[1,1,1,1,0,0,0,1,0,0]=>3 [.,[[.,[[.,.],.]],.]]=>[1,1,1,0,1,0,0,1,0,0]=>3 [.,[[[.,.],[.,.]],.]]=>[1,1,0,1,1,0,0,1,0,0]=>4 [.,[[[.,[.,.]],.],.]]=>[1,1,1,0,0,1,0,1,0,0]=>3 [.,[[[[.,.],.],.],.]]=>[1,1,0,1,0,1,0,1,0,0]=>4 [[.,.],[.,[.,[.,.]]]]=>[1,0,1,1,1,1,0,0,0,0]=>1 [[.,.],[.,[[.,.],.]]]=>[1,0,1,1,1,0,1,0,0,0]=>3 [[.,.],[[.,.],[.,.]]]=>[1,0,1,1,0,1,1,0,0,0]=>3 [[.,.],[[.,[.,.]],.]]=>[1,0,1,1,1,0,0,1,0,0]=>4 [[.,.],[[[.,.],.],.]]=>[1,0,1,1,0,1,0,1,0,0]=>4 [[.,[.,.]],[.,[.,.]]]=>[1,1,0,0,1,1,1,0,0,0]=>2 [[.,[.,.]],[[.,.],.]]=>[1,1,0,0,1,1,0,1,0,0]=>5 [[[.,.],.],[.,[.,.]]]=>[1,0,1,0,1,1,1,0,0,0]=>3 [[[.,.],.],[[.,.],.]]=>[1,0,1,0,1,1,0,1,0,0]=>6 [[.,[.,[.,.]]],[.,.]]=>[1,1,1,0,0,0,1,1,0,0]=>3 [[.,[[.,.],.]],[.,.]]=>[1,1,0,1,0,0,1,1,0,0]=>4 [[[.,.],[.,.]],[.,.]]=>[1,0,1,1,0,0,1,1,0,0]=>4 [[[.,[.,.]],.],[.,.]]=>[1,1,0,0,1,0,1,1,0,0]=>5 [[[[.,.],.],.],[.,.]]=>[1,0,1,0,1,0,1,1,0,0]=>6 [[.,[.,[.,[.,.]]]],.]=>[1,1,1,1,0,0,0,0,1,0]=>4 [[.,[.,[[.,.],.]]],.]=>[1,1,1,0,1,0,0,0,1,0]=>5 [[.,[[.,.],[.,.]]],.]=>[1,1,0,1,1,0,0,0,1,0]=>5 [[.,[[.,[.,.]],.]],.]=>[1,1,1,0,0,1,0,0,1,0]=>6 [[.,[[[.,.],.],.]],.]=>[1,1,0,1,0,1,0,0,1,0]=>6 [[[.,.],[.,[.,.]]],.]=>[1,0,1,1,1,0,0,0,1,0]=>5 [[[.,.],[[.,.],.]],.]=>[1,0,1,1,0,1,0,0,1,0]=>7 [[[.,[.,.]],[.,.]],.]=>[1,1,0,0,1,1,0,0,1,0]=>6 [[[[.,.],.],[.,.]],.]=>[1,0,1,0,1,1,0,0,1,0]=>7 [[[.,[.,[.,.]]],.],.]=>[1,1,1,0,0,0,1,0,1,0]=>7 [[[.,[[.,.],.]],.],.]=>[1,1,0,1,0,0,1,0,1,0]=>8 [[[[.,.],[.,.]],.],.]=>[1,0,1,1,0,0,1,0,1,0]=>8 [[[[.,[.,.]],.],.],.]=>[1,1,0,0,1,0,1,0,1,0]=>9 [[[[[.,.],.],.],.],.]=>[1,0,1,0,1,0,1,0,1,0]=>10 [.,[.,[.,[.,[.,[.,.]]]]]]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>0 [.,[.,[.,[.,[[.,.],.]]]]]=>[1,1,1,1,1,0,1,0,0,0,0,0]=>1 [.,[.,[.,[[.,.],[.,.]]]]]=>[1,1,1,1,0,1,1,0,0,0,0,0]=>1 [.,[.,[.,[[.,[.,.]],.]]]]=>[1,1,1,1,1,0,0,1,0,0,0,0]=>2 [.,[.,[.,[[[.,.],.],.]]]]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>2 [.,[.,[[.,.],[.,[.,.]]]]]=>[1,1,1,0,1,1,1,0,0,0,0,0]=>1 [.,[.,[[.,.],[[.,.],.]]]]=>[1,1,1,0,1,1,0,1,0,0,0,0]=>2 [.,[.,[[.,[.,.]],[.,.]]]]=>[1,1,1,1,0,0,1,1,0,0,0,0]=>2 [.,[.,[[[.,.],.],[.,.]]]]=>[1,1,1,0,1,0,1,1,0,0,0,0]=>2 [.,[.,[[.,[.,[.,.]]],.]]]=>[1,1,1,1,1,0,0,0,1,0,0,0]=>3 [.,[.,[[.,[[.,.],.]],.]]]=>[1,1,1,1,0,1,0,0,1,0,0,0]=>3 [.,[.,[[[.,.],[.,.]],.]]]=>[1,1,1,0,1,1,0,0,1,0,0,0]=>3 [.,[.,[[[.,[.,.]],.],.]]]=>[1,1,1,1,0,0,1,0,1,0,0,0]=>3 [.,[.,[[[[.,.],.],.],.]]]=>[1,1,1,0,1,0,1,0,1,0,0,0]=>3 [.,[[.,.],[.,[.,[.,.]]]]]=>[1,1,0,1,1,1,1,0,0,0,0,0]=>1 [.,[[.,.],[.,[[.,.],.]]]]=>[1,1,0,1,1,1,0,1,0,0,0,0]=>2 [.,[[.,.],[[.,.],[.,.]]]]=>[1,1,0,1,1,0,1,1,0,0,0,0]=>2 [.,[[.,.],[[.,[.,.]],.]]]=>[1,1,0,1,1,1,0,0,1,0,0,0]=>4 [.,[[.,.],[[[.,.],.],.]]]=>[1,1,0,1,1,0,1,0,1,0,0,0]=>4 [.,[[.,[.,.]],[.,[.,.]]]]=>[1,1,1,0,0,1,1,1,0,0,0,0]=>2 [.,[[.,[.,.]],[[.,.],.]]]=>[1,1,1,0,0,1,1,0,1,0,0,0]=>3 [.,[[[.,.],.],[.,[.,.]]]]=>[1,1,0,1,0,1,1,1,0,0,0,0]=>2 [.,[[[.,.],.],[[.,.],.]]]=>[1,1,0,1,0,1,1,0,1,0,0,0]=>4 [.,[[.,[.,[.,.]]],[.,.]]]=>[1,1,1,1,0,0,0,1,1,0,0,0]=>3 [.,[[.,[[.,.],.]],[.,.]]]=>[1,1,1,0,1,0,0,1,1,0,0,0]=>3 [.,[[[.,.],[.,.]],[.,.]]]=>[1,1,0,1,1,0,0,1,1,0,0,0]=>4 [.,[[[.,[.,.]],.],[.,.]]]=>[1,1,1,0,0,1,0,1,1,0,0,0]=>3 [.,[[[[.,.],.],.],[.,.]]]=>[1,1,0,1,0,1,0,1,1,0,0,0]=>4 [.,[[.,[.,[.,[.,.]]]],.]]=>[1,1,1,1,1,0,0,0,0,1,0,0]=>4 [.,[[.,[.,[[.,.],.]]],.]]=>[1,1,1,1,0,1,0,0,0,1,0,0]=>4 [.,[[.,[[.,.],[.,.]]],.]]=>[1,1,1,0,1,1,0,0,0,1,0,0]=>5 [.,[[.,[[.,[.,.]],.]],.]]=>[1,1,1,1,0,0,1,0,0,1,0,0]=>4 [.,[[.,[[[.,.],.],.]],.]]=>[1,1,1,0,1,0,1,0,0,1,0,0]=>5 [.,[[[.,.],[.,[.,.]]],.]]=>[1,1,0,1,1,1,0,0,0,1,0,0]=>5 [.,[[[.,.],[[.,.],.]],.]]=>[1,1,0,1,1,0,1,0,0,1,0,0]=>5 [.,[[[.,[.,.]],[.,.]],.]]=>[1,1,1,0,0,1,1,0,0,1,0,0]=>6 [.,[[[[.,.],.],[.,.]],.]]=>[1,1,0,1,0,1,1,0,0,1,0,0]=>6 [.,[[[.,[.,[.,.]]],.],.]]=>[1,1,1,1,0,0,0,1,0,1,0,0]=>4 [.,[[[.,[[.,.],.]],.],.]]=>[1,1,1,0,1,0,0,1,0,1,0,0]=>5 [.,[[[[.,.],[.,.]],.],.]]=>[1,1,0,1,1,0,0,1,0,1,0,0]=>5 [.,[[[[.,[.,.]],.],.],.]]=>[1,1,1,0,0,1,0,1,0,1,0,0]=>6 [.,[[[[[.,.],.],.],.],.]]=>[1,1,0,1,0,1,0,1,0,1,0,0]=>6 [[.,.],[.,[.,[.,[.,.]]]]]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>1 [[.,.],[.,[.,[[.,.],.]]]]=>[1,0,1,1,1,1,0,1,0,0,0,0]=>3 [[.,.],[.,[[.,.],[.,.]]]]=>[1,0,1,1,1,0,1,1,0,0,0,0]=>3 [[.,.],[.,[[.,[.,.]],.]]]=>[1,0,1,1,1,1,0,0,1,0,0,0]=>4 [[.,.],[.,[[[.,.],.],.]]]=>[1,0,1,1,1,0,1,0,1,0,0,0]=>4 [[.,.],[[.,.],[.,[.,.]]]]=>[1,0,1,1,0,1,1,1,0,0,0,0]=>3 [[.,.],[[.,.],[[.,.],.]]]=>[1,0,1,1,0,1,1,0,1,0,0,0]=>4 [[.,.],[[.,[.,.]],[.,.]]]=>[1,0,1,1,1,0,0,1,1,0,0,0]=>4 [[.,.],[[[.,.],.],[.,.]]]=>[1,0,1,1,0,1,0,1,1,0,0,0]=>4 [[.,.],[[.,[.,[.,.]]],.]]=>[1,0,1,1,1,1,0,0,0,1,0,0]=>5 [[.,.],[[.,[[.,.],.]],.]]=>[1,0,1,1,1,0,1,0,0,1,0,0]=>5 [[.,.],[[[.,.],[.,.]],.]]=>[1,0,1,1,0,1,1,0,0,1,0,0]=>7 [[.,.],[[[.,[.,.]],.],.]]=>[1,0,1,1,1,0,0,1,0,1,0,0]=>5 [[.,.],[[[[.,.],.],.],.]]=>[1,0,1,1,0,1,0,1,0,1,0,0]=>7 [[.,[.,.]],[.,[.,[.,.]]]]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>2 [[.,[.,.]],[.,[[.,.],.]]]=>[1,1,0,0,1,1,1,0,1,0,0,0]=>5 [[.,[.,.]],[[.,.],[.,.]]]=>[1,1,0,0,1,1,0,1,1,0,0,0]=>5 [[.,[.,.]],[[.,[.,.]],.]]=>[1,1,0,0,1,1,1,0,0,1,0,0]=>6 [[.,[.,.]],[[[.,.],.],.]]=>[1,1,0,0,1,1,0,1,0,1,0,0]=>6 [[[.,.],.],[.,[.,[.,.]]]]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>3 [[[.,.],.],[.,[[.,.],.]]]=>[1,0,1,0,1,1,1,0,1,0,0,0]=>6 [[[.,.],.],[[.,.],[.,.]]]=>[1,0,1,0,1,1,0,1,1,0,0,0]=>6 [[[.,.],.],[[.,[.,.]],.]]=>[1,0,1,0,1,1,1,0,0,1,0,0]=>7 [[[.,.],.],[[[.,.],.],.]]=>[1,0,1,0,1,1,0,1,0,1,0,0]=>7 [[.,[.,[.,.]]],[.,[.,.]]]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>3 [[.,[.,[.,.]]],[[.,.],.]]=>[1,1,1,0,0,0,1,1,0,1,0,0]=>7 [[.,[[.,.],.]],[.,[.,.]]]=>[1,1,0,1,0,0,1,1,1,0,0,0]=>4 [[.,[[.,.],.]],[[.,.],.]]=>[1,1,0,1,0,0,1,1,0,1,0,0]=>8 [[[.,.],[.,.]],[.,[.,.]]]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>4 [[[.,.],[.,.]],[[.,.],.]]=>[1,0,1,1,0,0,1,1,0,1,0,0]=>8 [[[.,[.,.]],.],[.,[.,.]]]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>5 [[[.,[.,.]],.],[[.,.],.]]=>[1,1,0,0,1,0,1,1,0,1,0,0]=>9 [[[[.,.],.],.],[.,[.,.]]]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>6 [[[[.,.],.],.],[[.,.],.]]=>[1,0,1,0,1,0,1,1,0,1,0,0]=>10 [[.,[.,[.,[.,.]]]],[.,.]]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>4 [[.,[.,[[.,.],.]]],[.,.]]=>[1,1,1,0,1,0,0,0,1,1,0,0]=>5 [[.,[[.,.],[.,.]]],[.,.]]=>[1,1,0,1,1,0,0,0,1,1,0,0]=>5 [[.,[[.,[.,.]],.]],[.,.]]=>[1,1,1,0,0,1,0,0,1,1,0,0]=>6 [[.,[[[.,.],.],.]],[.,.]]=>[1,1,0,1,0,1,0,0,1,1,0,0]=>6 [[[.,.],[.,[.,.]]],[.,.]]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>5 [[[.,.],[[.,.],.]],[.,.]]=>[1,0,1,1,0,1,0,0,1,1,0,0]=>7 [[[.,[.,.]],[.,.]],[.,.]]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>6 [[[[.,.],.],[.,.]],[.,.]]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>7 [[[.,[.,[.,.]]],.],[.,.]]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>7 [[[.,[[.,.],.]],.],[.,.]]=>[1,1,0,1,0,0,1,0,1,1,0,0]=>8 [[[[.,.],[.,.]],.],[.,.]]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>8 [[[[.,[.,.]],.],.],[.,.]]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>9 [[[[[.,.],.],.],.],[.,.]]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>10 [[.,[.,[.,[.,[.,.]]]]],.]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>5 [[.,[.,[.,[[.,.],.]]]],.]=>[1,1,1,1,0,1,0,0,0,0,1,0]=>6 [[.,[.,[[.,.],[.,.]]]],.]=>[1,1,1,0,1,1,0,0,0,0,1,0]=>6 [[.,[.,[[.,[.,.]],.]]],.]=>[1,1,1,1,0,0,1,0,0,0,1,0]=>7 [[.,[.,[[[.,.],.],.]]],.]=>[1,1,1,0,1,0,1,0,0,0,1,0]=>7 [[.,[[.,.],[.,[.,.]]]],.]=>[1,1,0,1,1,1,0,0,0,0,1,0]=>6 [[.,[[.,.],[[.,.],.]]],.]=>[1,1,0,1,1,0,1,0,0,0,1,0]=>7 [[.,[[.,[.,.]],[.,.]]],.]=>[1,1,1,0,0,1,1,0,0,0,1,0]=>7 [[.,[[[.,.],.],[.,.]]],.]=>[1,1,0,1,0,1,1,0,0,0,1,0]=>7 [[.,[[.,[.,[.,.]]],.]],.]=>[1,1,1,1,0,0,0,1,0,0,1,0]=>8 [[.,[[.,[[.,.],.]],.]],.]=>[1,1,1,0,1,0,0,1,0,0,1,0]=>8 [[.,[[[.,.],[.,.]],.]],.]=>[1,1,0,1,1,0,0,1,0,0,1,0]=>9 [[.,[[[.,[.,.]],.],.]],.]=>[1,1,1,0,0,1,0,1,0,0,1,0]=>8 [[.,[[[[.,.],.],.],.]],.]=>[1,1,0,1,0,1,0,1,0,0,1,0]=>9 [[[.,.],[.,[.,[.,.]]]],.]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>6 [[[.,.],[.,[[.,.],.]]],.]=>[1,0,1,1,1,0,1,0,0,0,1,0]=>8 [[[.,.],[[.,.],[.,.]]],.]=>[1,0,1,1,0,1,1,0,0,0,1,0]=>8 [[[.,.],[[.,[.,.]],.]],.]=>[1,0,1,1,1,0,0,1,0,0,1,0]=>9 [[[.,.],[[[.,.],.],.]],.]=>[1,0,1,1,0,1,0,1,0,0,1,0]=>9 [[[.,[.,.]],[.,[.,.]]],.]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>7 [[[.,[.,.]],[[.,.],.]],.]=>[1,1,0,0,1,1,0,1,0,0,1,0]=>10 [[[[.,.],.],[.,[.,.]]],.]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>8 [[[[.,.],.],[[.,.],.]],.]=>[1,0,1,0,1,1,0,1,0,0,1,0]=>11 [[[.,[.,[.,.]]],[.,.]],.]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>8 [[[.,[[.,.],.]],[.,.]],.]=>[1,1,0,1,0,0,1,1,0,0,1,0]=>9 [[[[.,.],[.,.]],[.,.]],.]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>9 [[[[.,[.,.]],.],[.,.]],.]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>10 [[[[[.,.],.],.],[.,.]],.]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>11 [[[.,[.,[.,[.,.]]]],.],.]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>9 [[[.,[.,[[.,.],.]]],.],.]=>[1,1,1,0,1,0,0,0,1,0,1,0]=>10 [[[.,[[.,.],[.,.]]],.],.]=>[1,1,0,1,1,0,0,0,1,0,1,0]=>10 [[[.,[[.,[.,.]],.]],.],.]=>[1,1,1,0,0,1,0,0,1,0,1,0]=>11 [[[.,[[[.,.],.],.]],.],.]=>[1,1,0,1,0,1,0,0,1,0,1,0]=>11 [[[[.,.],[.,[.,.]]],.],.]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>10 [[[[.,.],[[.,.],.]],.],.]=>[1,0,1,1,0,1,0,0,1,0,1,0]=>12 [[[[.,[.,.]],[.,.]],.],.]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>11 [[[[[.,.],.],[.,.]],.],.]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>12 [[[[.,[.,[.,.]]],.],.],.]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>12 [[[[.,[[.,.],.]],.],.],.]=>[1,1,0,1,0,0,1,0,1,0,1,0]=>13 [[[[[.,.],[.,.]],.],.],.]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>13 [[[[[.,[.,.]],.],.],.],.]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>14 [[[[[[.,.],.],.],.],.],.]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>15
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Description
The bounce statistic of a Dyck path.
The bounce path $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$.
The points where $D'$ touches the $x$-axis are called bounce points, and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$.
In particular, the bounce statistics of $D$ and $D'$ coincide.
Map
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
  • a leaf is associated with an empty Dyck path,
  • a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.