Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤ
Values
[1,0,1,0] => [1] => [1,0,1,0] => 1010 => 0
[1,0,1,0,1,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,0,1,1,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,0,0,1,0] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,1,0,1,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,0,1,0,1,0,1,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,0,1,1,0,0,1,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 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] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,0,0,1,0,1,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[1,1,0,1,0,0,1,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,1,0,1,1,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,0,0,0,1,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[1,1,1,0,0,1,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,1,1,0,1,0,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 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] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 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] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 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] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 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] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 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] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 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] => [1,0,1,1,0,0] => 101100 => 0
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
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searching the database for the individual values of this statistic
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Map
to binary word
Description
Return the Dyck word as binary word.
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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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