Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001594: Dyck paths ⟶ ℤ
Values
[1,0] => [1] => [.,.] => [1,0] => 1
[1,0,1,0] => [1,2] => [.,[.,.]] => [1,0,1,0] => 0
[1,1,0,0] => [2,1] => [[.,.],.] => [1,1,0,0] => 1
[1,0,1,0,1,0] => [1,2,3] => [.,[.,[.,.]]] => [1,0,1,0,1,0] => 0
[1,0,1,1,0,0] => [1,3,2] => [.,[[.,.],.]] => [1,0,1,1,0,0] => 0
[1,1,0,0,1,0] => [2,1,3] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[1,1,0,1,0,0] => [2,3,1] => [[.,[.,.]],.] => [1,1,0,1,0,0] => 0
[1,1,1,0,0,0] => [3,1,2] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => 0
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => 0
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 0
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 1
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 1
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 0
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,1,0,0] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,1,0,1,0,0] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => 1
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => 0
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => 0
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => 1
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => 0
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => 0
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => 0
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [.,[.,[[.,.],[.,[.,.]]]]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [.,[.,[[.,.],[[.,.],.]]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [.,[.,[[.,.],[.,[.,.]]]]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]] => [1,0,1,1,0,0,1,0,1,0,1,0] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [.,[[.,.],[[.,[.,.]],.]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]] => [1,0,1,1,0,1,0,1,0,1,0,0] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [.,[[.,.],[.,[.,[.,.]]]]] => [1,0,1,1,0,0,1,0,1,0,1,0] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [.,[[.,.],[.,[[.,.],.]]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [.,[[.,.],[[.,[.,.]],.]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 0
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Description
The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied.
See the link for the definition.
See the link for the definition.
Map
to Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
This is given recursively as follows:
- a leaf is associated to the empty Dyck Word
- a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
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