Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤ
Values
[1,0,1,0] => [1] => [1,0] => 10 => 0
[1,0,1,0,1,0] => [2,1] => [1,0,1,1,0,0] => 101100 => 0
[1,0,1,1,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,0,0,1,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,0,1,0,0] => [1] => [1,0] => 10 => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0
[1,0,1,1,0,1,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0
[1,1,0,0,1,0,1,0] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,1,0,0,0] => 111000 => 0
[1,1,0,1,0,0,1,0] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,0,1,1,0,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,1,0,0,0,1,0] => [3] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,0,0,1,0,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,1,0,1,0,0,0] => [1] => [1,0] => 10 => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 0
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,1,0,0,0] => 111000 => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,1,1,0,0,0,0,1,0] => [4] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0] => 10 => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => 111000 => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0] => 10 => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 0
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => 111000 => 0
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [1,0] => 10 => 0
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 0
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => 111000 => 0
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [1,0] => 10 => 0
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 0
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => 111000 => 0
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => [1,0] => 10 => 0
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => [1,0] => 10 => 0
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 0
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [1] => [1,0] => 10 => 0
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searching the database for the individual values of this statistic
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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 $(n-1,\ldots,2,1)$.
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