Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤ (values match St000932The number of occurrences of the pattern UDU in a Dyck path., St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.)
Values
[1,0] => [1] => [.,.] => [1,0] => 0
[1,0,1,0] => [2,1] => [[.,.],.] => [1,0,1,0] => 1
[1,1,0,0] => [1,2] => [.,[.,.]] => [1,1,0,0] => 0
[1,0,1,0,1,0] => [2,3,1] => [[.,.],[.,.]] => [1,0,1,1,0,0] => 1
[1,0,1,1,0,0] => [2,1,3] => [[.,.],[.,.]] => [1,0,1,1,0,0] => 1
[1,1,0,0,1,0] => [1,3,2] => [.,[[.,.],.]] => [1,1,0,1,0,0] => 1
[1,1,0,1,0,0] => [3,1,2] => [[.,[.,.]],.] => [1,1,0,0,1,0] => 0
[1,1,1,0,0,0] => [1,2,3] => [.,[.,[.,.]]] => [1,1,1,0,0,0] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 1
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 1
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 2
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 2
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 1
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 1
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 1
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 0
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 0
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 0
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => 0
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 2
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 2
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 2
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 2
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 2
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 2
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 2
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 0
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 0
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 0
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 0
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 1
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 1
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 0
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 0
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 0
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 0
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => 0
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [[.,.],[.,[.,[[.,.],.]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [[.,.],[.,[[.,[.,.]],.]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [[.,.],[.,[[.,[.,.]],.]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 1
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Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
Map
binary search tree: left to right
Description
Return the shape of the binary search tree of the permutation as a non labelled binary tree.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
The bijection is defined recursively as follows:
- a leaf is associated with an empty Dyck path,
- a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.
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