Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
St000947: Dyck paths ⟶ ℤ
Values
[1,1] => [1,0,1,0] => 1
[2] => [1,1,0,0] => 0
[1,1,1] => [1,0,1,0,1,0] => 3
[1,2] => [1,0,1,1,0,0] => 1
[2,1] => [1,1,0,0,1,0] => 2
[3] => [1,1,1,0,0,0] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => 6
[1,1,2] => [1,0,1,0,1,1,0,0] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => 4
[1,3] => [1,0,1,1,1,0,0,0] => 1
[2,1,1] => [1,1,0,0,1,0,1,0] => 5
[2,2] => [1,1,0,0,1,1,0,0] => 2
[3,1] => [1,1,1,0,0,0,1,0] => 3
[4] => [1,1,1,1,0,0,0,0] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5
[1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
[2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7
[3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
[5] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 10
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 8
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 8
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 14
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 5
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 21
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 15
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 16
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 10
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 17
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 11
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 12
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 6
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 18
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 12
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 13
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 7
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 14
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 8
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 9
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 3
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 19
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 13
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 14
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 8
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 15
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 9
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 10
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 4
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 16
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 10
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 11
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 5
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 12
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 7
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 20
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 14
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 15
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 9
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 16
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 10
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 11
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Description
The major index east count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The major index of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see St000027The major index of a Dyck path..
The major index east count is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The major index of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see St000027The major index of a Dyck path..
The major index east count is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Map
bounce path
Description
The bounce path determined by an integer composition.
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