Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001550: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,1] => [2,1] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => [2,3,1] => 0
[1,1,0,0,1,0] => [2,1,3] => [1,3,2] => 0
[1,1,0,1,0,0] => [2,3,1] => [2,1,3] => 0
[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [2,3,4,1] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,2,4,3] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [2,3,1,4] => 0
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [3,4,2,1] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [1,3,2,4] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,4,3,1] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,4,2,3] => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,1,4,3] => 0
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,2,1,4] => 0
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [1,4,3,2] => 0
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,1,3,4] => 0
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [3,2,4,1] => 0
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,1,2] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [2,3,4,1,5] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [3,4,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,2,4,3,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [2,3,5,4,1] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,2,5,3,4] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [2,3,1,5,4] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [3,4,2,1,5] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,2,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [2,3,1,4,5] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [3,4,2,5,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [4,5,3,1,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [1,3,2,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,4,3,5,1] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,4,3,1,5] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [3,5,4,2,1] => 1
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [1,4,2,3,5] => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [2,5,3,4,1] => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [1,5,2,3,4] => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,1,5,4,3] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [3,2,1,5,4] => 0
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [1,5,2,4,3] => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [2,1,4,5,3] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [3,2,5,1,4] => 0
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,3,1,2,5] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [1,4,3,2,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [2,5,4,3,1] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [1,5,3,2,4] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [2,1,5,3,4] => 0
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,2,1,4,5] => 0
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [1,5,4,2,3] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [2,1,4,3,5] => 0
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [3,2,5,4,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [4,3,1,5,2] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [1,5,4,3,2] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [2,1,3,4,5] => 0
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [3,2,4,5,1] => 0
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [4,3,5,1,2] => 0
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [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] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,4,6,5] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [2,3,4,5,1,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [3,4,5,6,2,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,3,5,4,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [2,3,4,6,5,1] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,3,6,4,5] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [2,3,4,1,6,5] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [3,4,5,2,1,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,3,6,5,4] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [2,3,4,1,5,6] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [3,4,5,2,6,1] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [4,5,6,3,1,2] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,2,4,3,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [2,3,5,4,6,1] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,2,4,3,6,5] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [2,3,5,4,1,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [3,4,6,5,2,1] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,2,5,3,4,6] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [2,3,6,4,5,1] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,2,6,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [2,3,1,6,5,4] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [3,4,2,1,6,5] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,2,6,3,5,4] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [2,3,1,5,6,4] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [3,4,2,6,1,5] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [4,5,3,1,2,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [2,3,6,5,4,1] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,2,6,4,3,5] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [2,3,1,6,4,5] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [3,4,2,1,5,6] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,2,6,5,3,4] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [2,3,1,5,4,6] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [3,4,2,6,5,1] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [4,5,3,1,6,2] => 0
>>> Load all 262 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 number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Map
to 312-avoiding permutation
Description
Map
ones to leading
Description
The unique permutation obtained by applying the Foata-Riordan map to obtain a Prüfer code, then prepending zero and cyclically shifting.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.
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!