Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001212: Dyck paths ⟶ ℤ
Values
[.,.] => [1,0] => [1,0] => 0
[.,[.,.]] => [1,1,0,0] => [1,0,1,0] => 1
[[.,.],.] => [1,0,1,0] => [1,1,0,0] => 0
[.,[.,[.,.]]] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 1
[.,[[.,.],.]] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 1
[[.,.],[.,.]] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
[[.,[.,.]],.] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 1
[[[.,.],.],.] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 1
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 2
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 1
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 1
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 1
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 1
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => 2
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 1
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 1
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 1
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 2
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 2
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 2
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 2
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => 2
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 1
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 1
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 1
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 2
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => 2
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 1
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 1
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 2
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 2
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 2
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 2
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 2
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 2
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 1
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 1
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 2
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 1
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 2
[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 2
[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 2
[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 2
[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 2
[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => 2
[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => 2
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 3
[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 2
[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 2
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 2
[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 2
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => 1
[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 1
[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 2
[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 1
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Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
Map
inverse Kreweras complement
Description
Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the i-th up step of the Dyck path is the size of the block of the set partition whose maximal element is i. If i is not a maximal element of a block, the (i+1)-st step is also an up step.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the i-th up step of the Dyck path is the size of the block of the set partition whose maximal element is i. If i is not a maximal element of a block, the (i+1)-st step is also an up step.
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)1T(r)0 where T(l) and T(r) are the images of this bijection to l and r.
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