- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001314: Dyck paths ⟶ ℤ

Values

[.,.] => [1,0] => 0
[.,[.,.]] => [1,1,0,0] => 0
[[.,.],.] => [1,0,1,0] => 0
[.,[.,[.,.]]] => [1,1,1,0,0,0] => 0
[.,[[.,.],.]] => [1,1,0,1,0,0] => 1
[[.,.],[.,.]] => [1,0,1,1,0,0] => 0
[[.,[.,.]],.] => [1,1,0,0,1,0] => 0
[[[.,.],.],.] => [1,0,1,0,1,0] => 0
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 0
[.,[.,[[.,.],.]]] => [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] => 2
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 0
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 1
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 0
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => 0
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => 0
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => 1
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => 0
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => 0
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => 0
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 0
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 3
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 5
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 5
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => 6
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 3
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 5
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 4
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 3
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => 3
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => 5
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => 4
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 3
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => 3
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 0
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 2
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 2
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 2
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => 2
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 0
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 1
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => 0
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => 1
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 0
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => 1
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => 0
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => 0
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => 0
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 0
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => 2
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => 2
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => 2
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => 2
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => 0
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => 1
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => 0
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => 0
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => 0
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => 1
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => 0
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => 0
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 2\ q + 4\ q^{2}$

$F_{5} = 16 + 5\ q + 8\ q^{2} + 6\ q^{3} + 2\ q^{4} + 4\ q^{5} + q^{6}$

Description

The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.

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 $r$ .