- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤ (values match St000015The number of peaks of a 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] => 0

[.,[.,.]] => [1,1,0,0] => 0

[[.,.],.] => [1,0,1,0] => 1

[.,[.,[.,.]]] => [1,1,1,0,0,0] => 0

[.,[[.,.],.]] => [1,1,0,1,0,0] => 1

[[.,.],[.,.]] => [1,0,1,1,0,0] => 1

[[.,[.,.]],.] => [1,1,0,0,1,0] => 1

[[[.,.],.],.] => [1,0,1,0,1,0] => 2

[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 0

[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 1

[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 1

[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => 1

[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => 2

[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 1

[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 2

[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 1

[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => 2

[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => 1

[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => 2

[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => 2

[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => 2

[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => 3

[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 0

[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 1

[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 1

[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 1

[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => 2

[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 1

[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 2

[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 1

[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 2

[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => 1

[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => 2

[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => 2

[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 2

[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => 3

[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 1

[[.,.],[.,[[.,.],.]]] => [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] => 3

[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 1

[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 2

[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => 2

[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => 3

[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 1

[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => 2

[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => 2

[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => 2

[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => 3

[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 1

[[.,[.,[[.,.],.]]],.] => [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] => 3

[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => 2

[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => 3

[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => 2

[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => 3

[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => 2

[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => 3

[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => 3

[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => 3

[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => 4

[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1

[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1

[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1

[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => 2

[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1

[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => 2

[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1

[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2

[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1

[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => 2

[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => 2

[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => 2

[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3

[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => 1

[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 2

[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 2

[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 2

[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => 3

[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => 1

[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => 2

[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 2

[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => 3

[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 1

[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => 2

[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 2

[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => 2

[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => 3

[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => 1

[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 2

[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 2

[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 2

[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => 3

[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2

[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => 3

[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 2

[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => 3

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + q^{2}$

$F_{4} = 1 + 6\ q + 6\ q^{2} + q^{3}$

$F_{5} = 1 + 10\ q + 20\ q^{2} + 10\ q^{3} + q^{4}$

$F_{6} = 1 + 15\ q + 50\ q^{2} + 50\ q^{3} + 15\ q^{4} + q^{5}$

$F_{7} = 1 + 21\ q + 105\ q^{2} + 175\ q^{3} + 105\ q^{4} + 21\ q^{5} + q^{6}$

Description

The number of valleys of the 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 $r$ .