Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
St000317: Permutations ⟶ ℤ
Values
[1,0] => [1] => 0
[1,0,1,0] => [1,2] => 0
[1,1,0,0] => [2,1] => 0
[1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => 0
[1,1,0,0,1,0] => [2,1,3] => 0
[1,1,0,1,0,0] => [2,3,1] => 0
[1,1,1,0,0,0] => [3,2,1] => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 0
[1,0,1,1,1,0,0,0] => [1,4,3,2] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0] => [2,4,3,1] => 0
[1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
[1,1,1,0,0,1,0,0] => [3,2,4,1] => 0
[1,1,1,0,1,0,0,0] => [3,4,2,1] => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 0
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 0
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 0
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 0
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 1
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 1
>>> Load all 196 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Map
to 312-avoiding permutation
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!