- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤ (values match St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2.)

Values

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

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

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

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

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

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

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

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

[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 6

[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 10

[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 9

[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 8

[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 8

[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => 7

[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 7

[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 6

[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 6

[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 5

[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => 7

[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => 6

[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => 5

[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 5

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

[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 6

[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 5

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

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

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

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

[[.,[.,.]],[[.,.],.]] => [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] => 2

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

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

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

[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 6

[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => 15

[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => 14

[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 13

[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => 13

[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => 12

[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => 12

[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => 11

[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 11

[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => 10

[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => 12

[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => 11

[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => 10

[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => 10

[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => 9

[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => 11

[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 10

[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 9

[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 9

[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => 8

[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => 9

[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => 8

[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 8

[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => 7

[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 9

[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => 8

[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 7

[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => 7

[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => 6

[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => 11

[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 10

[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 9

[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9

[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => 8

[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8

[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => 7

[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 7

[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => 6

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Description

The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with area sequences $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with St000005The bounce statistic of a Dyck path. and St000006The dinv of a Dyck path.. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.

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 $r$.