Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000542: Permutations ⟶ ℤ (values match St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.)
Values
[1,0] => [1] => [1] => [1] => 1
[1,0,1,0] => [2,1] => [1,2] => [2,1] => 2
[1,1,0,0] => [1,2] => [1,2] => [2,1] => 2
[1,0,1,0,1,0] => [3,2,1] => [1,2,3] => [3,2,1] => 3
[1,0,1,1,0,0] => [2,3,1] => [1,2,3] => [3,2,1] => 3
[1,1,0,0,1,0] => [3,1,2] => [1,2,3] => [3,2,1] => 3
[1,1,0,1,0,0] => [2,1,3] => [1,3,2] => [2,3,1] => 2
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 3
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 3
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 3
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 3
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 3
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [1,2,5,3,4] => [4,3,5,2,1] => 4
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [1,2,5,3,4] => [4,3,5,2,1] => 4
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => 3
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => 4
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => 4
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => 4
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => 4
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [1,5,2,4,3] => [3,4,2,5,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => 4
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [1,2,5,3,4] => [4,3,5,2,1] => 4
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [1,2,5,3,4] => [4,3,5,2,1] => 4
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [1,3,5,2,4] => [4,2,5,3,1] => 3
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => 3
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [1,2,4,5,3] => [3,5,4,2,1] => 3
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 5
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 5
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 5
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 5
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 5
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [1,2,6,3,5,4] => [4,5,3,6,2,1] => 4
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 5
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 5
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [1,2,5,6,3,4] => [4,3,6,5,2,1] => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [1,2,5,6,3,4] => [4,3,6,5,2,1] => 4
>>> 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 number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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].
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
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!