Identifier
-
Mp00141:
Binary trees
—pruning number to logarithmic height⟶
Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001550: Permutations ⟶ ℤ
Values
[.,.] => [1,0] => [1,0] => [1] => 0
[.,[.,.]] => [1,0,1,0] => [1,1,0,0] => [2,1] => 0
[[.,.],.] => [1,1,0,0] => [1,0,1,0] => [1,2] => 0
[.,[.,[.,.]]] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => 0
[.,[[.,.],.]] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => 0
[[.,.],[.,.]] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => [3,2,1] => 0
[[.,[.,.]],.] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,3,2] => 0
[[[.,.],.],.] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 0
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 0
[.,[[.,.],[.,.]]] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => 0
[.,[[.,[.,.]],.]] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
[.,[[[.,.],.],.]] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
[[.,.],[.,[.,.]]] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 0
[[.,.],[[.,.],.]] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
[[.,[.,.]],[.,.]] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
[[[.,.],.],[.,.]] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => 0
[[.,[.,[.,.]]],.] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 0
[[.,[[.,.],.]],.] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
[[[.,.],[.,.]],.] => [1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 0
[[[.,[.,.]],.],.] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 0
[[[[.,.],.],.],.] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0
[.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 0
[.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 0
[.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 0
[.,[[.,.],[.,[.,.]]]] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 0
[.,[[.,.],[[.,.],.]]] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 0
[.,[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 0
[.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 0
[.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 0
[.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 0
[.,[[[.,.],[.,.]],.]] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 0
[.,[[[.,[.,.]],.],.]] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 0
[[.,.],[.,[[.,.],.]]] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 0
[[.,.],[[.,.],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 0
[[.,.],[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
[[.,.],[[[.,.],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 0
[[.,[.,.]],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[[.,[.,.]],[[.,.],.]] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
[[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 0
[[[.,.],.],[[.,.],.]] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 0
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 0
[[.,[[.,.],.]],[.,.]] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 0
[[[.,.],[.,.]],[.,.]] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
[[[.,[.,.]],.],[.,.]] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 1
[[[[.,.],.],.],[.,.]] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 0
[[.,[.,[.,[.,.]]]],.] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 0
[[.,[.,[[.,.],.]]],.] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 0
[[.,[[.,.],[.,.]]],.] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 0
[[.,[[.,[.,.]],.]],.] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0
[[.,[[[.,.],.],.]],.] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
[[[.,.],[.,[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 0
[[[.,.],[[.,.],.]],.] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 0
[[[.,[.,.]],[.,.]],.] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 0
[[[[.,.],.],[.,.]],.] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 0
[[[.,[.,[.,.]]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 0
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0
[[[[.,.],[.,.]],.],.] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 0
[[[[.,[.,.]],.],.],.] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 0
[[[[[.,.],.],.],.],.] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 0
[.,[.,[.,[[.,.],[.,.]]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => 0
[.,[.,[.,[[[.,.],.],.]]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => 0
[.,[.,[[.,.],[.,[.,.]]]]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,4,1] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 0
[.,[.,[[.,[[.,.],.]],.]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => 0
[.,[.,[[[[.,.],.],.],.]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => 0
[.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => 0
[.,[[.,.],[[[.,.],.],.]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => 0
[.,[[[.,.],.],[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => 0
[.,[[[.,.],.],[[.,.],.]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,3,1,6] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,5,3,1] => 1
[.,[[[[.,.],.],.],[.,.]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,3,1] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 0
[.,[[.,[[.,.],[.,.]]],.]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 0
[.,[[.,[[[.,.],.],.]],.]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => 0
[.,[[[.,.],[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 0
[.,[[[[.,.],.],[.,.]],.]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,4,3] => 0
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Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Map
pruning number to logarithmic height
Description
Francon's map from binary trees to Dyck paths.
This bijection sends the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree., to the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path.. The implementation is a literal translation of Knuth's [2].
This bijection sends the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree., to the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path.. The implementation is a literal translation of Knuth's [2].
Map
to 312-avoiding permutation
Description
Map
reflect parallelogram polyomino
Description
Reflect the corresponding parallelogram polyomino, such that the first column becomes the first row.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
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