Identifier
-
Mp00038:
Integer compositions
—reverse⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤ
Values
[1,1] => [1,1] => [1,0,1,0] => 1
[2] => [2] => [1,1,0,0] => 0
[1,1,1] => [1,1,1] => [1,0,1,0,1,0] => 2
[1,2] => [2,1] => [1,1,0,0,1,0] => 2
[2,1] => [1,2] => [1,0,1,1,0,0] => 1
[3] => [3] => [1,1,1,0,0,0] => 0
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0] => 3
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
[1,3] => [3,1] => [1,1,1,0,0,0,1,0] => 3
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 2
[2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[3,1] => [1,3] => [1,0,1,1,1,0,0,0] => 1
[4] => [4] => [1,1,1,1,0,0,0,0] => 0
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 4
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 4
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 3
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 2
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
[5] => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 5
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 5
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 5
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 5
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 5
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 5
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 5
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 5
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 5
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
[2,1,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4
[2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[2,3,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 4
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
[3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[3,1,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
[3,2,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
[5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,1,1,1,1,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,1,1,1,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 6
[1,1,1,1,3] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 6
[1,1,1,2,1,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 6
[1,1,1,2,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 6
[1,1,1,3,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 6
[1,1,1,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 6
[1,1,2,1,1,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 6
[1,1,2,1,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 6
[1,1,2,2,1] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 6
[1,1,2,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 6
[1,1,3,1,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 6
[1,1,3,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 6
[1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 6
[1,1,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 6
[1,2,1,1,1,1] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 6
[1,2,1,1,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 6
[1,2,1,2,1] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 6
[1,2,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 6
[1,2,2,1,1] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 6
[1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 6
[1,2,3,1] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6
[1,2,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 6
[1,3,1,1,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 6
[1,3,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 6
[1,3,2,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
[1,3,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 6
[1,4,1,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 6
[1,4,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 6
[1,5,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[1,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
[2,1,1,1,1,1] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 5
[2,1,1,1,2] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 5
[2,1,1,2,1] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 5
[2,1,1,3] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 5
[2,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 5
[2,1,2,2] => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 5
[2,1,3,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 5
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Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$ \sum_v (j_v-i_v)/2. $$
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$ \sum_v (j_v-i_v)/2. $$
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
Map
bounce path
Description
The bounce path determined by an integer composition.
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