Identifier
-
Mp00295:
Standard tableaux
—valley composition⟶
Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000950: Dyck paths ⟶ ℤ
Values
[[1]] => [1] => [1] => [1,0] => 2
[[1,2]] => [2] => [1,1] => [1,0,1,0] => 2
[[1],[2]] => [2] => [1,1] => [1,0,1,0] => 2
[[1,2,3]] => [3] => [1,1,1] => [1,0,1,0,1,0] => 2
[[1,3],[2]] => [2,1] => [1,2] => [1,0,1,1,0,0] => 4
[[1,2],[3]] => [3] => [1,1,1] => [1,0,1,0,1,0] => 2
[[1],[2],[3]] => [3] => [1,1,1] => [1,0,1,0,1,0] => 2
[[1,2,3,4]] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 2
[[1,3,4],[2]] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
[[1,2,4],[3]] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 4
[[1,2,3],[4]] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 2
[[1,3],[2,4]] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
[[1,2],[3,4]] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 4
[[1,4],[2],[3]] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 4
[[1,3],[2],[4]] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
[[1,2],[3],[4]] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 2
[[1],[2],[3],[4]] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 2
[[1,2,3,4,5]] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 2
[[1,3,4,5],[2]] => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
[[1,2,4,5],[3]] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[[1,2,3,5],[4]] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[[1,2,3,4],[5]] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 2
[[1,3,5],[2,4]] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 8
[[1,2,5],[3,4]] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[[1,3,4],[2,5]] => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
[[1,2,4],[3,5]] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[[1,2,3],[4,5]] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[[1,4,5],[2],[3]] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[[1,3,5],[2],[4]] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 8
[[1,2,5],[3],[4]] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[[1,3,4],[2],[5]] => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
[[1,2,4],[3],[5]] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[[1,2,3],[4],[5]] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 2
[[1,4],[2,5],[3]] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[[1,3],[2,5],[4]] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 8
[[1,2],[3,5],[4]] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[[1,3],[2,4],[5]] => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
[[1,2],[3,4],[5]] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[[1,5],[2],[3],[4]] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
[[1,4],[2],[3],[5]] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[[1,3],[2],[4],[5]] => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
[[1,2],[3],[4],[5]] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 2
[[1],[2],[3],[4],[5]] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 2
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Description
Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1.
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
valley composition
Description
The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.
Map
bounce path
Description
The bounce path determined by an integer composition.
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