Identifier
-
Mp00102:
Dyck paths
—rise composition⟶
Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001024: Dyck paths ⟶ ℤ
Values
[1,0] => [1] => [1] => [1,0] => 1
[1,0,1,0] => [1,1] => [2] => [1,1,0,0] => 1
[1,1,0,0] => [2] => [1,1] => [1,0,1,0] => 2
[1,0,1,0,1,0] => [1,1,1] => [3] => [1,1,1,0,0,0] => 1
[1,0,1,1,0,0] => [1,2] => [2,1] => [1,1,0,0,1,0] => 1
[1,1,0,0,1,0] => [2,1] => [1,2] => [1,0,1,1,0,0] => 2
[1,1,0,1,0,0] => [2,1] => [1,2] => [1,0,1,1,0,0] => 2
[1,1,1,0,0,0] => [3] => [1,1,1] => [1,0,1,0,1,0] => 3
[1,0,1,0,1,0,1,0] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,0,0] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0] => 1
[1,0,1,1,0,0,1,0] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => 1
[1,0,1,1,0,1,0,0] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => 1
[1,0,1,1,1,0,0,0] => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
[1,1,0,0,1,0,1,0] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,0,1,1,0,0] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 2
[1,1,0,1,0,0,1,0] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,1,0,1,0,0] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,1,1,0,0,0] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 2
[1,1,1,0,0,0,1,0] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
[1,1,1,0,0,1,0,0] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
[1,1,1,0,1,0,0,0] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
[1,1,1,1,0,0,0,0] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,0,1,0,1,0,1,0,0] => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,0,1,0,1,1,0,0,0] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,1,0,1,1,0,0,0,1,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 3
[1,1,1,0,0,1,0,0,1,0] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,1,0,1,0,0] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 3
[1,1,1,0,1,0,0,0,1,0] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,0,1,0,0,1,0,0] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,0,1,0,1,0,0,0] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,0,1,1,0,0,0,0] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 3
[1,1,1,1,0,0,0,0,1,0] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,1,1,0,0,0,1,0,0] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,1,1,0,0,1,0,0,0] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,1,1,0,1,0,0,0,0] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,1,1,1,0,0,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
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Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Map
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
rise composition
Description
Send a Dyck path to the composition of sizes of its rises.
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