Identifier
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Images
=>
Cc0005;cc-rep-1Cc0010;cc-rep-2Cc0010;cc-rep-3
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[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]=>[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]] [1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[.,[.,[[[.,.],.],[.,[[.,.],.]]]]]=>[.,[.,[[[.,[.,[[.,.],.]]],.],.]]] [1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[.,[.,[[[.,.],.],[[.,.],[.,.]]]]]=>[.,[.,[[[.,[[.,[.,.]],.]],.],.]]] [1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[.,[.,[[[.,.],.],[[[.,.],.],.]]]]=>[.,[.,[[[.,[[[.,.],.],.]],.],.]]] [1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]=>[.,[.,[[[[.,[.,[.,.]]],.],.],.]]] [1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[.,[.,[[[[.,.],.],.],[[.,.],.]]]]=>[.,[.,[[[[.,[[.,.],.]],.],.],.]]] [1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[.,[.,[[[[[.,.],.],.],.],[.,.]]]]=>[.,[.,[[[[[.,[.,.]],.],.],.],.]]] [1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[[[[[[.,.],.],.],.],.],.]]]=>[.,[.,[[[[[[.,.],.],.],.],.],.]]] 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[1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] [1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] [1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]=>[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]] [1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]=>[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]] [1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]]=>[.,[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]] [1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]=>[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]] [1,1,1,1,2,3]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]]=>[.,[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]] [1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]=>[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]] [1,1,1,2,4]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[[.,.],[[[[.,.],.],.],.]]]]]=>[.,[.,[.,[[.,[[[[.,.],.],.],.]],.]]]] [1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]=>[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]] [1,1,2,5]=>[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[[.,.],[[[[[.,.],.],.],.],.]]]]=>[.,[.,[[.,[[[[[.,.],.],.],.],.]],.]]] [1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]=>[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]] [1,2,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]] [1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[[.,.],[[[[[[.,.],.],.],.],.],.]]]=>[.,[[.,[[[[[[.,.],.],.],.],.],.]],.]] [1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]=>[.,[[[[[[[[.,.],.],.],.],.],.],.],.]] [2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.] [2,2,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,[.,[.,[.,.]]]]]]]=>[[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]],.] [2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,.],[[[[[[[.,.],.],.],.],.],.],.]]=>[[.,[[[[[[[.,.],.],.],.],.],.],.]],.] [3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[.,.],.],[[[[[[.,.],.],.],.],.],.]]=>[[[.,[[[[[[.,.],.],.],.],.],.]],.],.] [8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]=>[[[[[[[[.,[.,.]],.],.],.],.],.],.],.] [9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],.]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],.] [1,1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] [1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]] [1,1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]] [1,1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]=>[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]] [1,1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]=>[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]] [1,1,1,1,6]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]=>[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]] [1,1,1,7]=>[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]=>[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]] [1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]=>[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]] [1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]=>[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]] [2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.] [2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[.,.],[[[[[[[[.,.],.],.],.],.],.],.],.]]=>[[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.] [9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]=>[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.] [10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to binary tree: up step, left tree, down step, right tree
Description
Return the binary tree corresponding to the Dyck path under the transformation up step - left tree - down step - right tree.
A Dyck path $D$ of semilength $n$ with $ n > 1$ may be uniquely decomposed into $1L0R$ for Dyck paths L,R of respective semilengths $n_1, n_2$ with $n_1 + n_2 = n-1$.
This map sends $D$ to the binary tree $T$ consisting of a root node with a left child according to $L$ and a right child according to $R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength $1$ is sent to a single node.
Map
left border symmetry
Description
Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees $T_1, T_2, T_3$ on its left border, it becomes $T_3, T_2, T_1$ where same symmetry has been applied to $T_1, T_2, T_3$.