Identifier
  • Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
    St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)
Values
[.,.] => [1,0] => 1
[.,[.,.]] => [1,1,0,0] => 1
[[.,.],.] => [1,0,1,0] => 2
[.,[.,[.,.]]] => [1,1,1,0,0,0] => 1
[.,[[.,.],.]] => [1,1,0,1,0,0] => 2
[[.,.],[.,.]] => [1,0,1,1,0,0] => 2
[[.,[.,.]],.] => [1,1,0,0,1,0] => 2
[[[.,.],.],.] => [1,0,1,0,1,0] => 3
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 1
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 2
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 2
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => 2
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => 3
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 2
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 3
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 2
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => 3
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => 2
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => 3
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => 3
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => 3
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => 4
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 1
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 2
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 2
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 2
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => 3
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 2
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 3
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 2
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 3
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => 2
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => 3
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => 3
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 3
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => 4
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 2
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 3
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 3
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 3
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => 4
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 2
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 3
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => 3
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => 4
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 2
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => 3
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => 3
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => 3
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => 4
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 2
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => 3
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => 3
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => 3
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => 4
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => 3
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => 4
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => 3
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => 4
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => 3
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => 4
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => 4
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => 4
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => 2
[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => 3
[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => 3
[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => 3
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => 3
[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => 4
[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 3
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 3
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 3
[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => 4
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => 2
[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => 3
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 3
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => 4
[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 2
[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => 3
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 3
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => 3
[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => 4
[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 3
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 3
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 3
[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => 4
[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => 3
[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => 4
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 3
[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => 4
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
[.,[[[.,[[.,.],.]],.],.]] => [1,1,1,0,1,0,0,1,0,1,0,0] => 4
[.,[[[[.,.],[.,.]],.],.]] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
[.,[[[[.,[.,.]],.],.],.]] => [1,1,1,0,0,1,0,1,0,1,0,0] => 4
[.,[[[[[.,.],.],.],.],.]] => [1,1,0,1,0,1,0,1,0,1,0,0] => 5
[[.,.],[.,[.,[.,[.,.]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[[.,.],[.,[.,[[.,.],.]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3
[[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 3
[[.,.],[.,[[.,[.,.]],.]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 3
[[.,.],[.,[[[.,.],.],.]]] => [1,0,1,1,1,0,1,0,1,0,0,0] => 4
[[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 3
[[.,.],[[.,.],[[.,.],.]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 4
[[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 3
[[.,.],[[[.,.],.],[.,.]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => 4
[[.,.],[[.,[.,[.,.]]],.]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 3
[[.,.],[[.,[[.,.],.]],.]] => [1,0,1,1,1,0,1,0,0,1,0,0] => 4
[[.,.],[[[.,.],[.,.]],.]] => [1,0,1,1,0,1,1,0,0,1,0,0] => 4
[[.,.],[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,0,1,0,1,0,0] => 4
[[.,.],[[[[.,.],.],.],.]] => [1,0,1,1,0,1,0,1,0,1,0,0] => 5
[[.,[.,.]],[.,[.,[.,.]]]] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
[[.,[.,.]],[.,[[.,.],.]]] => [1,1,0,0,1,1,1,0,1,0,0,0] => 3
[[.,[.,.]],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,1,1,0,0,0] => 3
[[.,[.,.]],[[.,[.,.]],.]] => [1,1,0,0,1,1,1,0,0,1,0,0] => 3
[[.,[.,.]],[[[.,.],.],.]] => [1,1,0,0,1,1,0,1,0,1,0,0] => 4
[[[.,.],.],[.,[.,[.,.]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
[[[.,.],.],[.,[[.,.],.]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4
[[[.,.],.],[[.,.],[.,.]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => 4
[[[.,.],.],[[.,[.,.]],.]] => [1,0,1,0,1,1,1,0,0,1,0,0] => 4
[[[.,.],.],[[[.,.],.],.]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 5
[[.,[.,[.,.]]],[.,[.,.]]] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2
[[.,[.,[.,.]]],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,1,0,0] => 3
[[.,[[.,.],.]],[.,[.,.]]] => [1,1,0,1,0,0,1,1,1,0,0,0] => 3
[[.,[[.,.],.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,1,0,0] => 4
[[[.,.],[.,.]],[.,[.,.]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[[[.,.],[.,.]],[[.,.],.]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 4
[[[.,[.,.]],.],[.,[.,.]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
[[[.,[.,.]],.],[[.,.],.]] => [1,1,0,0,1,0,1,1,0,1,0,0] => 4
[[[[.,.],.],.],[.,[.,.]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[[[[.,.],.],.],[[.,.],.]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 5
[[.,[.,[.,[.,.]]]],[.,.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[[.,[.,[[.,.],.]]],[.,.]] => [1,1,1,0,1,0,0,0,1,1,0,0] => 3
[[.,[[.,.],[.,.]]],[.,.]] => [1,1,0,1,1,0,0,0,1,1,0,0] => 3
[[.,[[.,[.,.]],.]],[.,.]] => [1,1,1,0,0,1,0,0,1,1,0,0] => 3
[[.,[[[.,.],.],.]],[.,.]] => [1,1,0,1,0,1,0,0,1,1,0,0] => 4
[[[.,.],[.,[.,.]]],[.,.]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
[[[.,.],[[.,.],.]],[.,.]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 4
[[[.,[.,.]],[.,.]],[.,.]] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3
[[[[.,.],.],[.,.]],[.,.]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[[[.,[.,[.,.]]],.],[.,.]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3
[[[.,[[.,.],.]],.],[.,.]] => [1,1,0,1,0,0,1,0,1,1,0,0] => 4
[[[[.,.],[.,.]],.],[.,.]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
[[[[.,[.,.]],.],.],[.,.]] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
[[[[[.,.],.],.],.],[.,.]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[.,[.,[[.,.],.]]]],.] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3
[[.,[.,[[.,.],[.,.]]]],.] => [1,1,1,0,1,1,0,0,0,0,1,0] => 3
[[.,[.,[[.,[.,.]],.]]],.] => [1,1,1,1,0,0,1,0,0,0,1,0] => 3
[[.,[.,[[[.,.],.],.]]],.] => [1,1,1,0,1,0,1,0,0,0,1,0] => 4
[[.,[[.,.],[.,[.,.]]]],.] => [1,1,0,1,1,1,0,0,0,0,1,0] => 3
[[.,[[.,.],[[.,.],.]]],.] => [1,1,0,1,1,0,1,0,0,0,1,0] => 4
[[.,[[.,[.,.]],[.,.]]],.] => [1,1,1,0,0,1,1,0,0,0,1,0] => 3
[[.,[[[.,.],.],[.,.]]],.] => [1,1,0,1,0,1,1,0,0,0,1,0] => 4
[[.,[[.,[.,[.,.]]],.]],.] => [1,1,1,1,0,0,0,1,0,0,1,0] => 3
[[.,[[.,[[.,.],.]],.]],.] => [1,1,1,0,1,0,0,1,0,0,1,0] => 4
[[.,[[[.,.],[.,.]],.]],.] => [1,1,0,1,1,0,0,1,0,0,1,0] => 4
[[.,[[[.,[.,.]],.],.]],.] => [1,1,1,0,0,1,0,1,0,0,1,0] => 4
[[.,[[[[.,.],.],.],.]],.] => [1,1,0,1,0,1,0,1,0,0,1,0] => 5
[[[.,.],[.,[.,[.,.]]]],.] => [1,0,1,1,1,1,0,0,0,0,1,0] => 3
[[[.,.],[.,[[.,.],.]]],.] => [1,0,1,1,1,0,1,0,0,0,1,0] => 4
[[[.,.],[[.,.],[.,.]]],.] => [1,0,1,1,0,1,1,0,0,0,1,0] => 4
[[[.,.],[[.,[.,.]],.]],.] => [1,0,1,1,1,0,0,1,0,0,1,0] => 4
[[[.,.],[[[.,.],.],.]],.] => [1,0,1,1,0,1,0,1,0,0,1,0] => 5
[[[.,[.,.]],[.,[.,.]]],.] => [1,1,0,0,1,1,1,0,0,0,1,0] => 3
[[[.,[.,.]],[[.,.],.]],.] => [1,1,0,0,1,1,0,1,0,0,1,0] => 4
[[[[.,.],.],[.,[.,.]]],.] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
[[[[.,.],.],[[.,.],.]],.] => [1,0,1,0,1,1,0,1,0,0,1,0] => 5
[[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[[[.,[[.,.],.]],[.,.]],.] => [1,1,0,1,0,0,1,1,0,0,1,0] => 4
[[[[.,.],[.,.]],[.,.]],.] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4
[[[[.,[.,.]],.],[.,.]],.] => [1,1,0,0,1,0,1,1,0,0,1,0] => 4
[[[[[.,.],.],.],[.,.]],.] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
[[[.,[.,[.,[.,.]]]],.],.] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3
[[[.,[.,[[.,.],.]]],.],.] => [1,1,1,0,1,0,0,0,1,0,1,0] => 4
[[[.,[[.,.],[.,.]]],.],.] => [1,1,0,1,1,0,0,0,1,0,1,0] => 4
[[[.,[[.,[.,.]],.]],.],.] => [1,1,1,0,0,1,0,0,1,0,1,0] => 4
[[[.,[[[.,.],.],.]],.],.] => [1,1,0,1,0,1,0,0,1,0,1,0] => 5
[[[[.,.],[.,[.,.]]],.],.] => [1,0,1,1,1,0,0,0,1,0,1,0] => 4
[[[[.,.],[[.,.],.]],.],.] => [1,0,1,1,0,1,0,0,1,0,1,0] => 5
[[[[.,[.,.]],[.,.]],.],.] => [1,1,0,0,1,1,0,0,1,0,1,0] => 4
[[[[[.,.],.],[.,.]],.],.] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
[[[[.,[.,[.,.]]],.],.],.] => [1,1,1,0,0,0,1,0,1,0,1,0] => 4
[[[[.,[[.,.],.]],.],.],.] => [1,1,0,1,0,0,1,0,1,0,1,0] => 5
[[[[[.,.],[.,.]],.],.],.] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5
[[[[[.,[.,.]],.],.],.],.] => [1,1,0,0,1,0,1,0,1,0,1,0] => 5
[[[[[[.,.],.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
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Description
The number of peaks of a Dyck path.
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:
  • 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 $r$.