Identifier
-
Mp00141:
Binary trees
—pruning number to logarithmic height⟶
Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: 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
[.,[[.,.],[.,.]]] => [1,0,1,1,1,0,0,0] => [1,1,1] => [[1],[2],[3]] => 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
[[.,.],[.,[.,.]]] => [1,1,1,0,0,0,1,0] => [3] => [[1,2,3]] => 0
[[.,.],[[.,.],.]] => [1,1,1,0,0,1,0,0] => [2] => [[1,2]] => 0
[[.,[.,.]],[.,.]] => [1,1,1,0,1,0,0,0] => [1] => [[1]] => 0
[[.,[.,[.,.]]],.] => [1,1,0,0,1,0,1,0] => [3,2] => [[1,2,3],[4,5]] => 0
[[.,[[.,.],.]],.] => [1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => 0
[[[.,.],[.,.]],.] => [1,1,0,1,1,0,0,0] => [1,1] => [[1],[2]] => 0
[[[.,[.,.]],.],.] => [1,1,0,1,0,0,1,0] => [3,1] => [[1,2,3],[4]] => 0
[[[[.,.],.],.],.] => [1,1,0,1,0,1,0,0] => [2,1] => [[1,2],[3]] => 0
[.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => 0
[.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [[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
[.,[[.,.],[[.,.],.]]] => [1,0,1,1,1,0,0,1,0,0] => [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
[.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 0
[.,[[[.,.],[.,.]],.]] => [1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => 0
[.,[[[.,[.,.]],.],.]] => [1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 0
[.,[[[[.,.],.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [[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
[[.,.],[[.,.],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => 0
[[.,.],[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => [[1,2,3,4],[5,6]] => 0
[[.,.],[[[.,.],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => [[1,2,3],[4,5]] => 0
[[.,[.,.]],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => [[1,2,3,4],[5]] => 0
[[.,[.,.]],[[.,.],.]] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,2,3],[4]] => 0
[[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => [4] => [[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
[[.,[.,[[.,.],.]]],.] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => 0
[[.,[[.,.],[.,.]]],.] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => 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
[[[.,.],[.,[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[1,2,3,4],[5],[6]] => 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
[[[.,[.,[.,.]]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [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
[[[[.,.],[.,.]],.],.] => [1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => 0
[[[[.,[.,.]],.],.],.] => [1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 0
[[[[[.,.],.],.],.],.] => [1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => 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
[.,[[[.,.],[.,.]],[.,.]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 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
[.,[[[[.,.],.],[.,.]],.]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [[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
[[.,.],[[[.,.],[.,.]],.]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [[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
[[.,[.,.]],[[[.,.],.],.]] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => [[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
[[[.,.],.],[[.,.],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [[1,2,3],[4,5,6]] => 0
[[[.,.],.],[[.,[.,.]],.]] => [1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => [[1,2,3,4,5],[6,7,8]] => 0
[[[.,.],.],[[[.,.],.],.]] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [[1,2,3,4],[5,6,7]] => 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
[[.,[.,[.,.]]],[[.,.],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 0
[[.,[[.,.],.]],[.,[.,.]]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [[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
[[[.,.],[.,.]],[.,[.,.]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [[1]] => 0
[[[.,[.,.]],.],[.,[.,.]]] => [1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [[1,2,3,4,5],[6,7]] => 0
[[[.,[.,.]],.],[[.,.],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[1,2,3,4],[5,6]] => 0
[[[[.,.],.],.],[.,[.,.]]] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[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
[[.,[.,[.,[.,.]]]],[.,.]] => [1,1,1,0,1,0,1,0,1,0,0,0] => [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
[[.,[[.,.],[.,.]]],[.,.]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => 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
[[[.,.],[.,[.,.]]],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[1,2,3,4,5]] => 0
[[[.,.],[[.,.],.]],[.,.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[1,2,3,4]] => 0
[[[.,[.,.]],[.,.]],[.,.]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[1,2,3]] => 0
[[[[.,.],.],[.,.]],[.,.]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[1,2]] => 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
[[[[.,.],[.,.]],.],[.,.]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1],[2]] => 0
[[[[.,[.,.]],.],.],[.,.]] => [1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,2,3],[4]] => 0
[[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[1,2],[3]] => 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
[[[.,.],[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,1,0,0,0] => [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
[[[[.,.],.],[.,[.,.]]],.] => [1,1,0,1,1,1,0,0,0,0,1,0] => [5,1,1,1] => [[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
>>> Load all 241 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The natural comajor index 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.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
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.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
Map
pruning number to logarithmic height
Description
Francon's map from binary trees to Dyck paths.
This bijection sends the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree., to the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path.. The implementation is a literal translation of Knuth's [2].
This bijection sends the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree., to the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path.. The implementation is a literal translation of Knuth's [2].
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!