Identifier
-
Mp00101:
Dyck paths
—decomposition reverse⟶
Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000240: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [1] => 1
[1,0,1,0] => [1,1,0,0] => [1,2] => 2
[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] => 3
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [3,1,2] => 3
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [1,3,2] => 2
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [2,1,3] => 2
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [2,3,1] => 1
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => 4
[1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => 4
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => 4
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => 4
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 3
[1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [3,1,4,2] => 3
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 3
[1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 3
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => 3
[1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 2
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 2
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 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] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => 5
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => 5
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => 5
[1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 5
[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => 5
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 5
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => 5
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => 5
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => 5
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 5
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 5
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 4
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => 4
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => 4
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 4
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => 4
[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 4
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => 4
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => 4
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => 4
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => 4
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => 4
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 3
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => 3
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 3
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => 3
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 3
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 3
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => 3
[1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 2
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 2
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 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] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,2,6,3,4,5] => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [5,1,6,2,3,4] => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,2,3,4,6] => 6
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => 6
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,5,6,2,3,4] => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => 6
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,6,3,4] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,2,6,3,5] => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,5,1,6,2,3] => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,2,5,3,4,6] => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [4,1,5,2,3,6] => 6
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,2,3,5,6] => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [3,1,2,4,5,6] => 6
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [3,6,1,2,4,5] => 6
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,6,2,3,5] => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,6,2,4,5] => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4,6] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [3,5,6,1,2,4] => 6
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,1,5,6,2,3] => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,6,3,5] => 6
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [3,1,2,6,4,5] => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [3,5,1,6,2,4] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4,6] => 6
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,1,2,5,6] => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [3,4,6,1,2,5] => 6
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searching the database for statistics with the same generating function
Description
The number of indices that are not small excedances.
A small excedance is an index $i$ for which $\pi_i = i+1$.
A small excedance is an index $i$ for which $\pi_i = i+1$.
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 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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