Identifier
-
Mp00101:
Dyck paths
—decomposition reverse⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000530: Permutations ⟶ ℤ
Values
[1,0,1,0] => [1,1,0,0] => [1,2] => 1
[1,1,0,0] => [1,0,1,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,2,3] => 1
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [2,1,3] => 2
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [3,1,2] => 2
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [2,3,1] => 2
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [3,2,1] => 1
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 3
[1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => 3
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => 5
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 3
[1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 3
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 5
[1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 5
[1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 3
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 5
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 9
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 6
[1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 6
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 9
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 9
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 16
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 6
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 16
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 11
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 4
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 4
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 6
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 6
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 16
[1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 4
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 9
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 11
[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 9
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 11
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 16
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 11
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 16
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 9
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 6
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 4
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 16
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 11
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 9
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 11
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 16
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 6
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 9
[1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 4
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 9
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 9
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,3,4,5,6] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,2,4,5,6] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [2,3,1,4,5,6] => 14
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,4,5,6] => 10
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,2,3,5,6] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,3,5,6] => 10
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [3,4,1,2,5,6] => 14
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,1,5,6] => 19
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,4,1,5,6] => 35
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,2,5,6] => 10
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,3,1,5,6] => 35
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,2,1,5,6] => 26
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,5,6] => 10
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,4,6] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,3,4,6] => 10
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,2,4,6] => 10
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,3,1,4,6] => 35
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,4,6] => 10
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [4,5,1,2,3,6] => 14
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [4,5,2,1,3,6] => 26
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [3,4,5,1,2,6] => 19
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,3,4,5,1,6] => 14
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,4,5,1,6] => 40
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [4,3,5,1,2,6] => 35
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [4,2,3,5,1,6] => 40
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [3,4,2,5,1,6] => 61
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,5,1,6] => 35
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,2,3,6] => 10
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1,3,6] => 10
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [5,3,4,1,2,6] => 35
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [5,2,3,4,1,6] => 40
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,4,1,6] => 35
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [4,5,3,1,2,6] => 26
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [4,5,2,3,1,6] => 61
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,2,1,6] => 26
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [4,3,5,2,1,6] => 40
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1,2,6] => 10
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Description
The number of permutations with the same descent word as the given permutation.
The descent word of a permutation is the binary word given by Mp00109descent word. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map Mp00109descent word containing a given permutation.
This statistic appears as up-down analysis in statistical applications in genetics, see [1] and the references therein.
The descent word of a permutation is the binary word given by Mp00109descent word. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map Mp00109descent word containing a given permutation.
This statistic appears as up-down analysis in statistical applications in genetics, see [1] and the references therein.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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