Identifier
-
Mp00012:
Binary trees
—to Dyck path: up step, left tree, down step, right tree⟶
Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤ
Values
[.,[.,.]] => [1,0,1,0] => [1] => [[1]] => 0
[.,[.,[.,.]]] => [1,0,1,0,1,0] => [2,1] => [[1,2],[3]] => 0
[.,[[.,.],.]] => [1,0,1,1,0,0] => [1,1] => [[1],[2]] => 0
[[.,.],[.,.]] => [1,1,0,0,1,0] => [2] => [[1,2]] => 0
[[.,[.,.]],.] => [1,1,0,1,0,0] => [1] => [[1]] => 0
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => 0
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => 0
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => [3,1,1] => [[1,2,3],[4],[5]] => 0
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => [2,1,1] => [[1,2],[3],[4]] => 0
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => [1,1,1] => [[1],[2],[3]] => 0
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => [3,2] => [[1,2,3],[4,5]] => 0
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => 0
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [3,1] => [[1,2,3],[4]] => 0
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [3] => [[1,2,3]] => 0
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => [2,1] => [[1,2],[3]] => 0
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => [1,1] => [[1],[2]] => 0
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [2] => [[1,2]] => 0
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => [1] => [[1]] => 0
[.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 0
[.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => 0
[.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 0
[.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 0
[.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 0
[.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 0
[.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => 0
[.,[[[.,.],[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 0
[.,[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[1,2],[3],[4],[5]] => 0
[.,[[[[.,.],.],.],.]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => 0
[[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => 0
[[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [[1,2,3,4],[5,6],[7,8]] => 0
[[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [[1,2,3],[4,5],[6,7]] => 0
[[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => 0
[[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [[1,2,3,4],[5,6,7],[8]] => 0
[[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [[1,2,3],[4,5,6],[7]] => 0
[[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => [[1,2,3,4],[5,6,7]] => 0
[[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => [[1,2,3],[4,5,6]] => 0
[[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 0
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[1,2,3,4],[5],[6]] => 0
[[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => [[1,2,3,4],[5,6]] => 0
[[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => [[1,2,3,4],[5]] => 0
[[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => [4] => [[1,2,3,4]] => 0
[[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => 0
[[.,[.,[[.,.],.]]],.] => [1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => 0
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[1,2,3],[4],[5]] => 0
[[.,[[.,[.,.]],.]],.] => [1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[1,2],[3],[4]] => 0
[[.,[[[.,.],.],.]],.] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => 0
[[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => [[1,2,3],[4,5]] => 0
[[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => 0
[[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,2,3],[4]] => 0
[[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => [3] => [[1,2,3]] => 0
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,1,0,1,0,0,0] => [2,1] => [[1,2],[3]] => 0
[[[.,[[.,.],.]],.],.] => [1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1],[2]] => 0
[[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => [2] => [[1,2]] => 0
[[[[.,[.,.]],.],.],.] => [1,1,1,1,0,1,0,0,0,0] => [1] => [[1]] => 0
[.,[[.,[[[.,.],.],.]],.]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 0
[.,[[[[.,.],.],[.,.]],.]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 0
[.,[[[.,[.,[.,.]]],.],.]] => [1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 0
[.,[[[.,[[.,.],.]],.],.]] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => 0
[.,[[[[.,.],[.,.]],.],.]] => [1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 0
[.,[[[[.,[.,.]],.],.],.]] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => 0
[.,[[[[[.,.],.],.],.],.]] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 0
[[.,.],[[[[.,.],.],.],.]] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 0
[[[[.,.],.],.],[[.,.],.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => [[1,2,3,4],[5,6,7,8]] => 0
[[.,[[[.,.],.],.]],[.,.]] => [1,1,0,1,1,1,0,0,0,0,1,0] => [5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => 0
[[[[.,.],.],[.,.]],[.,.]] => [1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => [[1,2,3,4,5],[6,7,8]] => 0
[[[.,[.,[.,.]]],.],[.,.]] => [1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,1] => [[1,2,3,4,5],[6,7],[8]] => 0
[[[.,[[.,.],.]],.],[.,.]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 0
[[[[.,.],[.,.]],.],[.,.]] => [1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [[1,2,3,4,5],[6,7]] => 0
[[[[.,[.,.]],.],.],[.,.]] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[1,2,3,4,5],[6]] => 0
[[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[1,2,3,4,5]] => 0
[[.,[.,[[.,[.,.]],.]]],.] => [1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 0
[[.,[.,[[[.,.],.],.]]],.] => [1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => 0
[[.,[[.,.],[[.,.],.]]],.] => [1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 0
[[.,[[.,[.,.]],[.,.]]],.] => [1,1,0,1,1,0,1,0,0,1,0,0] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 0
[[.,[[[.,.],.],[.,.]]],.] => [1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 0
[[.,[[.,[.,[.,.]]],.]],.] => [1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 0
[[.,[[.,[[.,.],.]],.]],.] => [1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => 0
[[.,[[[.,.],[.,.]],.]],.] => [1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 0
[[.,[[[.,[.,.]],.],.]],.] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[1,2],[3],[4],[5]] => 0
[[.,[[[[.,.],.],.],.]],.] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => 0
[[[.,.],[.,[[.,.],.]]],.] => [1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => 0
[[[.,.],[[.,.],[.,.]]],.] => [1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => [[1,2,3,4],[5,6],[7,8]] => 0
[[[.,.],[[.,[.,.]],.]],.] => [1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [[1,2,3],[4,5],[6,7]] => 0
[[[.,.],[[[.,.],.],.]],.] => [1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => 0
[[[.,[.,.]],[.,[.,.]]],.] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => [[1,2,3,4],[5,6,7],[8]] => 0
[[[.,[.,.]],[[.,.],.]],.] => [1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [[1,2,3],[4,5,6],[7]] => 0
[[[[.,.],.],[.,[.,.]]],.] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [[1,2,3,4],[5,6,7]] => 0
[[[[.,.],.],[[.,.],.]],.] => [1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [[1,2,3],[4,5,6]] => 0
[[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 0
[[[.,[[.,.],.]],[.,.]],.] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [[1,2,3,4],[5],[6]] => 0
[[[[.,.],[.,.]],[.,.]],.] => [1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[1,2,3,4],[5,6]] => 0
[[[[.,[.,.]],.],[.,.]],.] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[1,2,3,4],[5]] => 0
[[[[[.,.],.],.],[.,.]],.] => [1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[1,2,3,4]] => 0
[[[.,[.,[.,[.,.]]]],.],.] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => 0
[[[.,[.,[[.,.],.]]],.],.] => [1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => 0
[[[.,[[.,.],[.,.]]],.],.] => [1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[1,2,3],[4],[5]] => 0
[[[.,[[.,[.,.]],.]],.],.] => [1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[1,2],[3],[4]] => 0
[[[.,[[[.,.],.],.]],.],.] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => 0
[[[[.,.],[.,[.,.]]],.],.] => [1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[1,2,3],[4,5]] => 0
[[[[.,.],[[.,.],.]],.],.] => [1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[1,2],[3,4]] => 0
[[[[.,[.,.]],[.,.]],.],.] => [1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,2,3],[4]] => 0
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searching the database for the individual values of this statistic
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Map
to Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
This is given recursively as follows:
- a leaf is associated to the empty Dyck Word
- a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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