Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000201: Binary trees ⟶ ℤ (values match St000196The number of occurrences of the contiguous pattern [[.,.],[.,.)
Values
[1] => [1,0] => [.,.] => 1
[2] => [1,0,1,0] => [.,[.,.]] => 1
[1,1] => [1,1,0,0] => [[.,.],.] => 1
[3] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => 1
[2,1] => [1,0,1,1,0,0] => [.,[[.,.],.]] => 1
[1,1,1] => [1,1,0,1,0,0] => [[[.,.],.],.] => 1
[4] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => 1
[2,2] => [1,1,1,0,0,0] => [[.,.],[.,.]] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [.,[[[.,.],.],.]] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [[[[.,.],.],.],.] => 1
[5] => [1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [.,[.,[[[.,.],.],.]]] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [[.,.],[[.,.],.]] => 2
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [[[[[.,.],.],.],.],.] => 1
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,.]]]]]] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],.]]]]] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [.,[.,[[.,.],[.,.]]]] => 2
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[[[.,.],.],.]]]] => 1
[3,3] => [1,1,1,0,1,0,0,0] => [[.,[.,.]],[.,.]] => 2
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [.,[[.,.],[[.,.],.]]] => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[[[[.,.],.],.],.]]] => 1
[2,2,2] => [1,1,1,1,0,0,0,0] => [[[.,.],.],[.,.]] => 2
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,.],.],.]] => 2
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [.,[[[[[.,.],.],.],.],.]] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[[[[[.,.],.],.],.],.],.] => 1
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => 1
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[[.,.],.]]]]]] => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[[.,.],[.,.]]]]] => 2
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[.,[[[.,.],.],.]]]]] => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [.,[[.,[.,.]],[.,.]]] => 2
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[[.,.],[[.,.],.]]]] => 2
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[.,[[[[.,.],.],.],.]]]] => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [[.,[.,.]],[[.,.],.]] => 2
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [.,[[[.,.],.],[.,.]]] => 2
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [.,[[.,.],[[[.,.],.],.]]] => 2
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [.,[.,[[[[[.,.],.],.],.],.]]] => 1
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [[[.,.],.],[[.,.],.]] => 2
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[.,.],[[[[.,.],.],.],.]] => 2
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [.,[[[[[[.,.],.],.],.],.],.]] => 1
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[[[[[[.,.],.],.],.],.],.],.] => 1
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[[.,.],[.,.]]]]]] => 2
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[[.,[.,.]],[.,.]]]] => 2
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[.,[[.,.],[[.,.],.]]]]] => 2
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [[.,[.,[.,.]]],[.,.]] => 2
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [.,[[.,[.,.]],[[.,.],.]]] => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[[[.,.],.],[.,.]]]] => 2
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [.,[.,[[.,.],[[[.,.],.],.]]]] => 2
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [[.,[[.,.],.]],[.,.]] => 2
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [[.,[.,.]],[[[.,.],.],.]] => 2
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [.,[[[.,.],.],[[.,.],.]]] => 2
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [.,[[.,.],[[[[.,.],.],.],.]]] => 2
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[.,.],.],.],[.,.]] => 2
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [[[.,.],.],[[[.,.],.],.]] => 2
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [[.,.],[[[[[.,.],.],.],.],.]] => 2
[2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [.,[[[[[[[.,.],.],.],.],.],.],.]] => 1
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[[[[[[[.,.],.],.],.],.],.],.],.] => 1
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[.,[[.,[.,.]],[.,.]]]]] => 2
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [.,[[.,[.,[.,.]]],[.,.]]] => 2
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [.,[.,[[.,[.,.]],[[.,.],.]]]] => 2
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[.,[[[.,.],.],[.,.]]]]] => 2
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [[.,[.,[.,.]]],[[.,.],.]] => 2
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [.,[[.,[[.,.],.]],[.,.]]] => 2
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [.,[[.,[.,.]],[[[.,.],.],.]]] => 2
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [.,[.,[[[.,.],.],[[.,.],.]]]] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,.]] => 3
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[.,[[.,.],.]],[[.,.],.]] => 2
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [[.,[.,.]],[[[[.,.],.],.],.]] => 2
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [.,[[[[.,.],.],.],[.,.]]] => 2
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [.,[[[.,.],.],[[[.,.],.],.]]] => 2
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => 2
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [[[.,.],.],[[[[.,.],.],.],.]] => 2
[2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[.,.],[[[[[[.,.],.],.],.],.],.]] => 2
[2,1,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [.,[[[[[[[[.,.],.],.],.],.],.],.],.]] => 1
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [.,[.,[[.,[.,[.,.]]],[.,.]]]] => 2
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [[.,[.,[.,[.,.]]]],[.,.]] => 2
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [.,[[.,[.,[.,.]]],[[.,.],.]]] => 2
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [.,[.,[[.,[[.,.],.]],[.,.]]]] => 2
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [[.,[.,[[.,.],.]]],[.,.]] => 2
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [[.,[.,[.,.]]],[[[.,.],.],.]] => 2
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [.,[[[.,.],[.,.]],[.,.]]] => 3
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [.,[[.,[[.,.],.]],[[.,.],.]]] => 2
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [.,[.,[[[[.,.],.],.],[.,.]]]] => 2
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [[[.,.],[[.,.],.]],[.,.]] => 3
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [[.,[[[.,.],.],.]],[.,.]] => 2
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [[.,[[.,.],.]],[[[.,.],.],.]] => 2
[3,3,1,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [[.,[.,.]],[[[[[.,.],.],.],.],.]] => 2
[3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [.,[[[[.,.],.],.],[[.,.],.]]] => 2
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [[[[[.,.],.],.],.],[.,.]] => 2
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [[[[.,.],.],.],[[[.,.],.],.]] => 2
[2,1,1,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]] => 1
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [.,[[.,[.,[.,[.,.]]]],[.,.]]] => 2
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [[.,[.,[.,[.,.]]]],[[.,.],.]] => 2
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [.,[[.,[.,[[.,.],.]]],[.,.]]] => 2
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [.,[.,[[[.,.],[.,.]],[.,.]]]] => 3
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [[.,[[.,.],[.,.]]],[.,.]] => 3
[4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [[.,[.,[[.,.],.]]],[[.,.],.]] => 2
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [.,[[[.,.],[[.,.],.]],[.,.]]] => 3
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Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
logarithmic height to pruning number
Description
Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].
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