Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001256: Dyck paths ⟶ ℤ
Values
[.,.] => [1,0] => [1,0] => 1
[.,[.,.]] => [1,1,0,0] => [1,0,1,0] => 1
[[.,.],.] => [1,0,1,0] => [1,1,0,0] => 1
[.,[.,[.,.]]] => [1,1,1,0,0,0] => [1,1,0,0,1,0] => 1
[.,[[.,.],.]] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 1
[[.,.],[.,.]] => [1,0,1,1,0,0] => [1,1,1,0,0,0] => 1
[[.,[.,.]],.] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 1
[[[.,.],.],.] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 1
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => 1
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,0,1,0] => 1
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => 1
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,0] => 1
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 1
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 1
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 1
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => 1
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => 1
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,0] => 1
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => 1
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 1
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 1
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 1
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => 2
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 1
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 1
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 1
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 1
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 1
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 1
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 1
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 1
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 1
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 1
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => 1
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => 1
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 1
[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 1
[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => 1
[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 1
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 1
[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 1
[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 1
[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
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Description
Number of simple reflexive modules that are 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
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.
Map
inverse promotion
Description
The inverse promotion of a Dyck path.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.
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