Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤ
Values
[1,0,1,0] => [1] => [1,0,1,0] => 1010 => 1
[1,0,1,0,1,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[1,0,1,1,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[1,1,0,0,1,0] => [2] => [1,1,0,0,1,0] => 110010 => 1
[1,1,0,1,0,0] => [1] => [1,0,1,0] => 1010 => 1
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[1,1,0,1,1,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[1,1,1,0,0,1,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 1
[1,1,1,0,1,0,0,0] => [1] => [1,0,1,0] => 1010 => 1
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 1
[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 => 1
[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 => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 1
[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 => 1
[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 => 1
[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 => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 1
[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 => 1
[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 => 1
[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 => 1
[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 => 1
[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 => 1
[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 => 1
[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 => 1
[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 => 1
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 1
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Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
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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