Identifier
-
Mp00029:
Dyck paths
—to binary tree: left tree, up step, right tree, down step⟶
Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000030: Permutations ⟶ ℤ
Values
[1,0] => [.,.] => [1] => [1] => 0
[1,0,1,0] => [[.,.],.] => [1,2] => [2,1] => 1
[1,1,0,0] => [.,[.,.]] => [2,1] => [1,2] => 0
[1,0,1,0,1,0] => [[[.,.],.],.] => [1,2,3] => [3,2,1] => 2
[1,0,1,1,0,0] => [[.,.],[.,.]] => [3,1,2] => [2,1,3] => 1
[1,1,0,0,1,0] => [[.,[.,.]],.] => [2,1,3] => [3,1,2] => 2
[1,1,0,1,0,0] => [.,[[.,.],.]] => [2,3,1] => [1,3,2] => 1
[1,1,1,0,0,0] => [.,[.,[.,.]]] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [[[[.,.],.],.],.] => [1,2,3,4] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0] => [[[.,.],.],[.,.]] => [4,1,2,3] => [3,2,1,4] => 2
[1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => [3,1,2,4] => [4,2,1,3] => 3
[1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => [3,4,1,2] => [2,1,4,3] => 2
[1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => [4,3,1,2] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => [2,1,3,4] => [4,3,1,2] => 3
[1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => [4,2,1,3] => [3,1,2,4] => 2
[1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => [2,3,1,4] => [4,1,3,2] => 4
[1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => [2,3,4,1] => [1,4,3,2] => 2
[1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => [4,2,3,1] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => [3,2,1,4] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => [3,2,4,1] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => [3,4,2,1] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [4,3,2,1,5] => 3
[1,0,1,0,1,1,0,0,1,0] => [[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [5,3,2,1,4] => 4
[1,0,1,0,1,1,0,1,0,0] => [[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [3,2,1,5,4] => 3
[1,0,1,0,1,1,1,0,0,0] => [[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [3,2,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0] => [[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [5,4,2,1,3] => 4
[1,0,1,1,0,0,1,1,0,0] => [[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [4,2,1,3,5] => 3
[1,0,1,1,0,1,0,0,1,0] => [[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [5,2,1,4,3] => 5
[1,0,1,1,0,1,0,1,0,0] => [[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [2,1,5,4,3] => 3
[1,0,1,1,0,1,1,0,0,0] => [[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [2,1,4,3,5] => 2
[1,0,1,1,1,0,0,0,1,0] => [[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [5,2,1,3,4] => 4
[1,0,1,1,1,0,0,1,0,0] => [[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0] => [[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [2,1,3,5,4] => 2
[1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [5,4,3,1,2] => 4
[1,1,0,0,1,0,1,1,0,0] => [[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [4,3,1,2,5] => 3
[1,1,0,0,1,1,0,0,1,0] => [[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [5,3,1,2,4] => 4
[1,1,0,0,1,1,0,1,0,0] => [[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [3,1,2,5,4] => 3
[1,1,0,0,1,1,1,0,0,0] => [[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [3,1,2,4,5] => 2
[1,1,0,1,0,0,1,0,1,0] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [5,4,1,3,2] => 5
[1,1,0,1,0,0,1,1,0,0] => [[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [4,1,3,2,5] => 4
[1,1,0,1,0,1,0,0,1,0] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [5,1,4,3,2] => 6
[1,1,0,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,5,4,3,2] => 3
[1,1,0,1,0,1,1,0,0,0] => [.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [1,4,3,2,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [5,1,3,2,4] => 5
[1,1,0,1,1,0,0,1,0,0] => [.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [1,5,3,2,4] => 3
[1,1,0,1,1,0,1,0,0,0] => [.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [1,3,2,5,4] => 2
[1,1,0,1,1,1,0,0,0,0] => [.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [5,4,1,2,3] => 4
[1,1,1,0,0,0,1,1,0,0] => [[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [4,1,2,3,5] => 3
[1,1,1,0,0,1,0,0,1,0] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [5,1,4,2,3] => 6
[1,1,1,0,0,1,0,1,0,0] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,5,4,2,3] => 3
[1,1,1,0,0,1,1,0,0,0] => [.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [1,4,2,3,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [5,1,2,4,3] => 5
[1,1,1,0,1,0,0,1,0,0] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,5,2,4,3] => 4
[1,1,1,0,1,0,1,0,0,0] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,5,4,3] => 2
[1,1,1,0,1,1,0,0,0,0] => [.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [1,2,4,3,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [5,1,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,5,2,3,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,2,5,3,4] => 2
[1,1,1,1,0,1,0,0,0,0] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 4
[1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => [6,4,3,2,1,5] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => [[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => [6,5,3,2,1,4] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => [5,3,2,1,4,6] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => [6,3,2,1,5,4] => 6
[1,0,1,0,1,1,0,1,0,1,0,0] => [[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => [[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => [6,3,2,1,4,5] => 5
[1,0,1,0,1,1,1,0,0,1,0,0] => [[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => [3,2,1,6,4,5] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => [3,2,1,4,6,5] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => [6,5,4,2,1,3] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => [5,4,2,1,3,6] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => [6,4,2,1,3,5] => 5
[1,0,1,1,0,0,1,1,0,1,0,0] => [[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => [4,2,1,3,6,5] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => [4,2,1,3,5,6] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => [6,5,2,1,4,3] => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => [[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => [5,2,1,4,3,6] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => [6,2,1,5,4,3] => 7
[1,0,1,1,0,1,0,1,0,1,0,0] => [[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => [2,1,6,5,4,3] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => [2,1,5,4,3,6] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => [6,2,1,4,3,5] => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => [[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => [2,1,6,4,3,5] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => [6,5,2,1,3,4] => 5
[1,0,1,1,1,0,0,0,1,1,0,0] => [[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => [5,2,1,3,4,6] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => [6,2,1,5,3,4] => 7
[1,0,1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => [2,1,6,5,3,4] => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => [2,1,5,3,4,6] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => [6,2,1,3,5,4] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => [2,1,6,3,5,4] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => [2,1,3,6,5,4] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => [2,1,3,5,4,6] => 2
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Description
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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.
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.
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.
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.
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