Identifier
Values
=>
Cc0005;cc-rep-0 Cc0010;cc-rep
[1,0,1,0]=>[[.,.],.]=>1 [1,1,0,0]=>[.,[.,.]]=>2 [1,0,1,0,1,0]=>[[[.,.],.],.]=>1 [1,0,1,1,0,0]=>[[.,.],[.,.]]=>2 [1,1,0,0,1,0]=>[[.,[.,.]],.]=>2 [1,1,0,1,0,0]=>[.,[[.,.],.]]=>3 [1,1,1,0,0,0]=>[.,[.,[.,.]]]=>5 [1,0,1,0,1,0,1,0]=>[[[[.,.],.],.],.]=>1 [1,0,1,0,1,1,0,0]=>[[[.,.],.],[.,.]]=>2 [1,0,1,1,0,0,1,0]=>[[[.,.],[.,.]],.]=>2 [1,0,1,1,0,1,0,0]=>[[.,.],[[.,.],.]]=>3 [1,0,1,1,1,0,0,0]=>[[.,.],[.,[.,.]]]=>5 [1,1,0,0,1,0,1,0]=>[[[.,[.,.]],.],.]=>2 [1,1,0,0,1,1,0,0]=>[[.,[.,.]],[.,.]]=>4 [1,1,0,1,0,0,1,0]=>[[.,[[.,.],.]],.]=>3 [1,1,0,1,0,1,0,0]=>[.,[[[.,.],.],.]]=>4 [1,1,0,1,1,0,0,0]=>[.,[[.,.],[.,.]]]=>7 [1,1,1,0,0,0,1,0]=>[[.,[.,[.,.]]],.]=>5 [1,1,1,0,0,1,0,0]=>[.,[[.,[.,.]],.]]=>7 [1,1,1,0,1,0,0,0]=>[.,[.,[[.,.],.]]]=>9 [1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,.]]]]=>14 [1,0,1,0,1,0,1,0,1,0]=>[[[[[.,.],.],.],.],.]=>1 [1,0,1,0,1,0,1,1,0,0]=>[[[[.,.],.],.],[.,.]]=>2 [1,0,1,0,1,1,0,0,1,0]=>[[[[.,.],.],[.,.]],.]=>2 [1,0,1,0,1,1,0,1,0,0]=>[[[.,.],.],[[.,.],.]]=>3 [1,0,1,0,1,1,1,0,0,0]=>[[[.,.],.],[.,[.,.]]]=>5 [1,0,1,1,0,0,1,0,1,0]=>[[[[.,.],[.,.]],.],.]=>2 [1,0,1,1,0,0,1,1,0,0]=>[[[.,.],[.,.]],[.,.]]=>4 [1,0,1,1,0,1,0,0,1,0]=>[[[.,.],[[.,.],.]],.]=>3 [1,0,1,1,0,1,0,1,0,0]=>[[.,.],[[[.,.],.],.]]=>4 [1,0,1,1,0,1,1,0,0,0]=>[[.,.],[[.,.],[.,.]]]=>7 [1,0,1,1,1,0,0,0,1,0]=>[[[.,.],[.,[.,.]]],.]=>5 [1,0,1,1,1,0,0,1,0,0]=>[[.,.],[[.,[.,.]],.]]=>7 [1,0,1,1,1,0,1,0,0,0]=>[[.,.],[.,[[.,.],.]]]=>9 [1,0,1,1,1,1,0,0,0,0]=>[[.,.],[.,[.,[.,.]]]]=>14 [1,1,0,0,1,0,1,0,1,0]=>[[[[.,[.,.]],.],.],.]=>2 [1,1,0,0,1,0,1,1,0,0]=>[[[.,[.,.]],.],[.,.]]=>4 [1,1,0,0,1,1,0,0,1,0]=>[[[.,[.,.]],[.,.]],.]=>4 [1,1,0,0,1,1,0,1,0,0]=>[[.,[.,.]],[[.,.],.]]=>6 [1,1,0,0,1,1,1,0,0,0]=>[[.,[.,.]],[.,[.,.]]]=>10 [1,1,0,1,0,0,1,0,1,0]=>[[[.,[[.,.],.]],.],.]=>3 [1,1,0,1,0,0,1,1,0,0]=>[[.,[[.,.],.]],[.,.]]=>6 [1,1,0,1,0,1,0,0,1,0]=>[[.,[[[.,.],.],.]],.]=>4 [1,1,0,1,0,1,0,1,0,0]=>[.,[[[[.,.],.],.],.]]=>5 [1,1,0,1,0,1,1,0,0,0]=>[.,[[[.,.],.],[.,.]]]=>9 [1,1,0,1,1,0,0,0,1,0]=>[[.,[[.,.],[.,.]]],.]=>7 [1,1,0,1,1,0,0,1,0,0]=>[.,[[[.,.],[.,.]],.]]=>9 [1,1,0,1,1,0,1,0,0,0]=>[.,[[.,.],[[.,.],.]]]=>12 [1,1,0,1,1,1,0,0,0,0]=>[.,[[.,.],[.,[.,.]]]]=>19 [1,1,1,0,0,0,1,0,1,0]=>[[[.,[.,[.,.]]],.],.]=>5 [1,1,1,0,0,0,1,1,0,0]=>[[.,[.,[.,.]]],[.,.]]=>10 [1,1,1,0,0,1,0,0,1,0]=>[[.,[[.,[.,.]],.]],.]=>7 [1,1,1,0,0,1,0,1,0,0]=>[.,[[[.,[.,.]],.],.]]=>9 [1,1,1,0,0,1,1,0,0,0]=>[.,[[.,[.,.]],[.,.]]]=>16 [1,1,1,0,1,0,0,0,1,0]=>[[.,[.,[[.,.],.]]],.]=>9 [1,1,1,0,1,0,0,1,0,0]=>[.,[[.,[[.,.],.]],.]]=>12 [1,1,1,0,1,0,1,0,0,0]=>[.,[.,[[[.,.],.],.]]]=>14 [1,1,1,0,1,1,0,0,0,0]=>[.,[.,[[.,.],[.,.]]]]=>23 [1,1,1,1,0,0,0,0,1,0]=>[[.,[.,[.,[.,.]]]],.]=>14 [1,1,1,1,0,0,0,1,0,0]=>[.,[[.,[.,[.,.]]],.]]=>19 [1,1,1,1,0,0,1,0,0,0]=>[.,[.,[[.,[.,.]],.]]]=>23 [1,1,1,1,0,1,0,0,0,0]=>[.,[.,[.,[[.,.],.]]]]=>28 [1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[.,.]]]]]=>42 [1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[.,.],.],.],.],.],.]=>1 [1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[.,.],.],.],.],[.,.]]=>2 [1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[.,.],.],.],[.,.]],.]=>2 [1,0,1,0,1,0,1,1,0,1,0,0]=>[[[[.,.],.],.],[[.,.],.]]=>3 [1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[.,.],.],.],[.,[.,.]]]=>5 [1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[.,.],.],[.,.]],.],.]=>2 [1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[.,.],.],[.,.]],[.,.]]=>4 [1,0,1,0,1,1,0,1,0,0,1,0]=>[[[[.,.],.],[[.,.],.]],.]=>3 [1,0,1,0,1,1,0,1,0,1,0,0]=>[[[.,.],.],[[[.,.],.],.]]=>4 [1,0,1,0,1,1,0,1,1,0,0,0]=>[[[.,.],.],[[.,.],[.,.]]]=>7 [1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[.,.],.],[.,[.,.]]],.]=>5 [1,0,1,0,1,1,1,0,0,1,0,0]=>[[[.,.],.],[[.,[.,.]],.]]=>7 [1,0,1,0,1,1,1,0,1,0,0,0]=>[[[.,.],.],[.,[[.,.],.]]]=>9 [1,0,1,0,1,1,1,1,0,0,0,0]=>[[[.,.],.],[.,[.,[.,.]]]]=>14 [1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[.,.],[.,.]],.],.],.]=>2 [1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[.,.],[.,.]],.],[.,.]]=>4 [1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[.,.],[.,.]],[.,.]],.]=>4 [1,0,1,1,0,0,1,1,0,1,0,0]=>[[[.,.],[.,.]],[[.,.],.]]=>6 [1,0,1,1,0,0,1,1,1,0,0,0]=>[[[.,.],[.,.]],[.,[.,.]]]=>10 [1,0,1,1,0,1,0,0,1,0,1,0]=>[[[[.,.],[[.,.],.]],.],.]=>3 [1,0,1,1,0,1,0,0,1,1,0,0]=>[[[.,.],[[.,.],.]],[.,.]]=>6 [1,0,1,1,0,1,0,1,0,0,1,0]=>[[[.,.],[[[.,.],.],.]],.]=>4 [1,0,1,1,0,1,0,1,0,1,0,0]=>[[.,.],[[[[.,.],.],.],.]]=>5 [1,0,1,1,0,1,0,1,1,0,0,0]=>[[.,.],[[[.,.],.],[.,.]]]=>9 [1,0,1,1,0,1,1,0,0,0,1,0]=>[[[.,.],[[.,.],[.,.]]],.]=>7 [1,0,1,1,0,1,1,0,0,1,0,0]=>[[.,.],[[[.,.],[.,.]],.]]=>9 [1,0,1,1,0,1,1,0,1,0,0,0]=>[[.,.],[[.,.],[[.,.],.]]]=>12 [1,0,1,1,0,1,1,1,0,0,0,0]=>[[.,.],[[.,.],[.,[.,.]]]]=>19 [1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[.,.],[.,[.,.]]],.],.]=>5 [1,0,1,1,1,0,0,0,1,1,0,0]=>[[[.,.],[.,[.,.]]],[.,.]]=>10 [1,0,1,1,1,0,0,1,0,0,1,0]=>[[[.,.],[[.,[.,.]],.]],.]=>7 [1,0,1,1,1,0,0,1,0,1,0,0]=>[[.,.],[[[.,[.,.]],.],.]]=>9 [1,0,1,1,1,0,0,1,1,0,0,0]=>[[.,.],[[.,[.,.]],[.,.]]]=>16 [1,0,1,1,1,0,1,0,0,0,1,0]=>[[[.,.],[.,[[.,.],.]]],.]=>9 [1,0,1,1,1,0,1,0,0,1,0,0]=>[[.,.],[[.,[[.,.],.]],.]]=>12 [1,0,1,1,1,0,1,0,1,0,0,0]=>[[.,.],[.,[[[.,.],.],.]]]=>14 [1,0,1,1,1,0,1,1,0,0,0,0]=>[[.,.],[.,[[.,.],[.,.]]]]=>23 [1,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,.]]]],.]=>14 [1,0,1,1,1,1,0,0,0,1,0,0]=>[[.,.],[[.,[.,[.,.]]],.]]=>19 [1,0,1,1,1,1,0,0,1,0,0,0]=>[[.,.],[.,[[.,[.,.]],.]]]=>23 [1,0,1,1,1,1,0,1,0,0,0,0]=>[[.,.],[.,[.,[[.,.],.]]]]=>28 [1,0,1,1,1,1,1,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,.]]]]]=>42 [1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[.,[.,.]],.],.],.],.]=>2 [1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[.,[.,.]],.],.],[.,.]]=>4 [1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[.,[.,.]],.],[.,.]],.]=>4 [1,1,0,0,1,0,1,1,0,1,0,0]=>[[[.,[.,.]],.],[[.,.],.]]=>6 [1,1,0,0,1,0,1,1,1,0,0,0]=>[[[.,[.,.]],.],[.,[.,.]]]=>10 [1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[.,[.,.]],[.,.]],.],.]=>4 [1,1,0,0,1,1,0,0,1,1,0,0]=>[[[.,[.,.]],[.,.]],[.,.]]=>8 [1,1,0,0,1,1,0,1,0,0,1,0]=>[[[.,[.,.]],[[.,.],.]],.]=>6 [1,1,0,0,1,1,0,1,0,1,0,0]=>[[.,[.,.]],[[[.,.],.],.]]=>8 [1,1,0,0,1,1,0,1,1,0,0,0]=>[[.,[.,.]],[[.,.],[.,.]]]=>14 [1,1,0,0,1,1,1,0,0,0,1,0]=>[[[.,[.,.]],[.,[.,.]]],.]=>10 [1,1,0,0,1,1,1,0,0,1,0,0]=>[[.,[.,.]],[[.,[.,.]],.]]=>14 [1,1,0,0,1,1,1,0,1,0,0,0]=>[[.,[.,.]],[.,[[.,.],.]]]=>18 [1,1,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,.]]]]=>28 [1,1,0,1,0,0,1,0,1,0,1,0]=>[[[[.,[[.,.],.]],.],.],.]=>3 [1,1,0,1,0,0,1,0,1,1,0,0]=>[[[.,[[.,.],.]],.],[.,.]]=>6 [1,1,0,1,0,0,1,1,0,0,1,0]=>[[[.,[[.,.],.]],[.,.]],.]=>6 [1,1,0,1,0,0,1,1,0,1,0,0]=>[[.,[[.,.],.]],[[.,.],.]]=>9 [1,1,0,1,0,0,1,1,1,0,0,0]=>[[.,[[.,.],.]],[.,[.,.]]]=>15 [1,1,0,1,0,1,0,0,1,0,1,0]=>[[[.,[[[.,.],.],.]],.],.]=>4 [1,1,0,1,0,1,0,0,1,1,0,0]=>[[.,[[[.,.],.],.]],[.,.]]=>8 [1,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[[[[.,.],.],.],.]],.]=>5 [1,1,0,1,0,1,0,1,0,1,0,0]=>[.,[[[[[.,.],.],.],.],.]]=>6 [1,1,0,1,0,1,0,1,1,0,0,0]=>[.,[[[[.,.],.],.],[.,.]]]=>11 [1,1,0,1,0,1,1,0,0,0,1,0]=>[[.,[[[.,.],.],[.,.]]],.]=>9 [1,1,0,1,0,1,1,0,0,1,0,0]=>[.,[[[[.,.],.],[.,.]],.]]=>11 [1,1,0,1,0,1,1,0,1,0,0,0]=>[.,[[[.,.],.],[[.,.],.]]]=>15 [1,1,0,1,0,1,1,1,0,0,0,0]=>[.,[[[.,.],.],[.,[.,.]]]]=>24 [1,1,0,1,1,0,0,0,1,0,1,0]=>[[[.,[[.,.],[.,.]]],.],.]=>7 [1,1,0,1,1,0,0,0,1,1,0,0]=>[[.,[[.,.],[.,.]]],[.,.]]=>14 [1,1,0,1,1,0,0,1,0,0,1,0]=>[[.,[[[.,.],[.,.]],.]],.]=>9 [1,1,0,1,1,0,0,1,0,1,0,0]=>[.,[[[[.,.],[.,.]],.],.]]=>11 [1,1,0,1,1,0,0,1,1,0,0,0]=>[.,[[[.,.],[.,.]],[.,.]]]=>20 [1,1,0,1,1,0,1,0,0,0,1,0]=>[[.,[[.,.],[[.,.],.]]],.]=>12 [1,1,0,1,1,0,1,0,0,1,0,0]=>[.,[[[.,.],[[.,.],.]],.]]=>15 [1,1,0,1,1,0,1,0,1,0,0,0]=>[.,[[.,.],[[[.,.],.],.]]]=>18 [1,1,0,1,1,0,1,1,0,0,0,0]=>[.,[[.,.],[[.,.],[.,.]]]]=>30 [1,1,0,1,1,1,0,0,0,0,1,0]=>[[.,[[.,.],[.,[.,.]]]],.]=>19 [1,1,0,1,1,1,0,0,0,1,0,0]=>[.,[[[.,.],[.,[.,.]]],.]]=>24 [1,1,0,1,1,1,0,0,1,0,0,0]=>[.,[[.,.],[[.,[.,.]],.]]]=>30 [1,1,0,1,1,1,0,1,0,0,0,0]=>[.,[[.,.],[.,[[.,.],.]]]]=>37 [1,1,0,1,1,1,1,0,0,0,0,0]=>[.,[[.,.],[.,[.,[.,.]]]]]=>56 [1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[.,[.,[.,.]]],.],.],.]=>5 [1,1,1,0,0,0,1,0,1,1,0,0]=>[[[.,[.,[.,.]]],.],[.,.]]=>10 [1,1,1,0,0,0,1,1,0,0,1,0]=>[[[.,[.,[.,.]]],[.,.]],.]=>10 [1,1,1,0,0,0,1,1,0,1,0,0]=>[[.,[.,[.,.]]],[[.,.],.]]=>15 [1,1,1,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,.]]],[.,[.,.]]]=>25 [1,1,1,0,0,1,0,0,1,0,1,0]=>[[[.,[[.,[.,.]],.]],.],.]=>7 [1,1,1,0,0,1,0,0,1,1,0,0]=>[[.,[[.,[.,.]],.]],[.,.]]=>14 [1,1,1,0,0,1,0,1,0,0,1,0]=>[[.,[[[.,[.,.]],.],.]],.]=>9 [1,1,1,0,0,1,0,1,0,1,0,0]=>[.,[[[[.,[.,.]],.],.],.]]=>11 [1,1,1,0,0,1,0,1,1,0,0,0]=>[.,[[[.,[.,.]],.],[.,.]]]=>20 [1,1,1,0,0,1,1,0,0,0,1,0]=>[[.,[[.,[.,.]],[.,.]]],.]=>16 [1,1,1,0,0,1,1,0,0,1,0,0]=>[.,[[[.,[.,.]],[.,.]],.]]=>20 [1,1,1,0,0,1,1,0,1,0,0,0]=>[.,[[.,[.,.]],[[.,.],.]]]=>27 [1,1,1,0,0,1,1,1,0,0,0,0]=>[.,[[.,[.,.]],[.,[.,.]]]]=>43 [1,1,1,0,1,0,0,0,1,0,1,0]=>[[[.,[.,[[.,.],.]]],.],.]=>9 [1,1,1,0,1,0,0,0,1,1,0,0]=>[[.,[.,[[.,.],.]]],[.,.]]=>18 [1,1,1,0,1,0,0,1,0,0,1,0]=>[[.,[[.,[[.,.],.]],.]],.]=>12 [1,1,1,0,1,0,0,1,0,1,0,0]=>[.,[[[.,[[.,.],.]],.],.]]=>15 [1,1,1,0,1,0,0,1,1,0,0,0]=>[.,[[.,[[.,.],.]],[.,.]]]=>27 [1,1,1,0,1,0,1,0,0,0,1,0]=>[[.,[.,[[[.,.],.],.]]],.]=>14 [1,1,1,0,1,0,1,0,0,1,0,0]=>[.,[[.,[[[.,.],.],.]],.]]=>18 [1,1,1,0,1,0,1,0,1,0,0,0]=>[.,[.,[[[[.,.],.],.],.]]]=>20 [1,1,1,0,1,0,1,1,0,0,0,0]=>[.,[.,[[[.,.],.],[.,.]]]]=>34 [1,1,1,0,1,1,0,0,0,0,1,0]=>[[.,[.,[[.,.],[.,.]]]],.]=>23 [1,1,1,0,1,1,0,0,0,1,0,0]=>[.,[[.,[[.,.],[.,.]]],.]]=>30 [1,1,1,0,1,1,0,0,1,0,0,0]=>[.,[.,[[[.,.],[.,.]],.]]]=>34 [1,1,1,0,1,1,0,1,0,0,0,0]=>[.,[.,[[.,.],[[.,.],.]]]]=>43 [1,1,1,0,1,1,1,0,0,0,0,0]=>[.,[.,[[.,.],[.,[.,.]]]]]=>66 [1,1,1,1,0,0,0,0,1,0,1,0]=>[[[.,[.,[.,[.,.]]]],.],.]=>14 [1,1,1,1,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,.]]]],[.,.]]=>28 [1,1,1,1,0,0,0,1,0,0,1,0]=>[[.,[[.,[.,[.,.]]],.]],.]=>19 [1,1,1,1,0,0,0,1,0,1,0,0]=>[.,[[[.,[.,[.,.]]],.],.]]=>24 [1,1,1,1,0,0,0,1,1,0,0,0]=>[.,[[.,[.,[.,.]]],[.,.]]]=>43 [1,1,1,1,0,0,1,0,0,0,1,0]=>[[.,[.,[[.,[.,.]],.]]],.]=>23 [1,1,1,1,0,0,1,0,0,1,0,0]=>[.,[[.,[[.,[.,.]],.]],.]]=>30 [1,1,1,1,0,0,1,0,1,0,0,0]=>[.,[.,[[[.,[.,.]],.],.]]]=>34 [1,1,1,1,0,0,1,1,0,0,0,0]=>[.,[.,[[.,[.,.]],[.,.]]]]=>57 [1,1,1,1,0,1,0,0,0,0,1,0]=>[[.,[.,[.,[[.,.],.]]]],.]=>28 [1,1,1,1,0,1,0,0,0,1,0,0]=>[.,[[.,[.,[[.,.],.]]],.]]=>37 [1,1,1,1,0,1,0,0,1,0,0,0]=>[.,[.,[[.,[[.,.],.]],.]]]=>43 [1,1,1,1,0,1,0,1,0,0,0,0]=>[.,[.,[.,[[[.,.],.],.]]]]=>48 [1,1,1,1,0,1,1,0,0,0,0,0]=>[.,[.,[.,[[.,.],[.,.]]]]]=>76 [1,1,1,1,1,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],.]=>42 [1,1,1,1,1,0,0,0,0,1,0,0]=>[.,[[.,[.,[.,[.,.]]]],.]]=>56 [1,1,1,1,1,0,0,0,1,0,0,0]=>[.,[.,[[.,[.,[.,.]]],.]]]=>66 [1,1,1,1,1,0,0,1,0,0,0,0]=>[.,[.,[.,[[.,[.,.]],.]]]]=>76 [1,1,1,1,1,0,1,0,0,0,0,0]=>[.,[.,[.,[.,[[.,.],.]]]]]=>90 [1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,.]]]]]]=>132
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Description
The number of elements smaller than a binary tree in Tamari order.
Map
to binary tree: left tree, up step, right tree, down step
Description
Return the binary tree corresponding to the Dyck path under the transformation left tree - up step - right tree - down step.
A Dyck path $D$ of semilength $n$ with $n > 1$ may be uniquely decomposed into $L 1 R 0$ 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.
This map may also be described as the unique map sending the Tamari orders on Dyck paths to the Tamari order on binary trees.