Identifier
Values
[1,0] => [2,1] => {{1,2}} => [2,1] => 1
[1,0,1,0] => [3,1,2] => {{1,3},{2}} => [3,2,1] => 2
[1,1,0,0] => [2,3,1] => {{1,2,3}} => [2,3,1] => 2
[1,0,1,0,1,0] => [4,1,2,3] => {{1,4},{2},{3}} => [4,2,3,1] => 3
[1,0,1,1,0,0] => [3,1,4,2] => {{1,3,4},{2}} => [3,2,4,1] => 3
[1,1,0,0,1,0] => [2,4,1,3] => {{1,2,4},{3}} => [2,4,3,1] => 3
[1,1,0,1,0,0] => [4,3,1,2] => {{1,4},{2,3}} => [4,3,2,1] => 2
[1,1,1,0,0,0] => [2,3,4,1] => {{1,2,3,4}} => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 4
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 4
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 3
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => {{1,3,4,5},{2}} => [3,2,4,5,1] => 4
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 4
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => {{1,2,4,5},{3}} => [2,4,3,5,1] => 4
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 3
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => {{1,4,5},{2,3}} => [4,3,2,5,1] => 3
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => {{1,2,3,5},{4}} => [2,3,5,4,1] => 4
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => {{1,2,5},{3,4}} => [2,5,4,3,1] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => {{1,5},{2,3,4}} => [5,3,4,2,1] => 3
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => 5
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => 5
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => 5
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}} => [6,2,3,5,4,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 5
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => 5
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => 5
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}} => [6,2,4,3,5,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}} => [6,2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => 4
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => 5
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}} => [3,2,6,5,4,1] => 4
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}} => [6,2,4,5,3,1] => 4
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 5
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => 5
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}} => [2,5,3,4,6,1] => 5
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 5
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}} => [2,6,3,5,4,1] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 5
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}} => [6,3,2,4,5,1] => 4
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}} => [5,3,2,4,6,1] => 4
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}} => [6,4,3,2,5,1] => 4
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}} => [5,6,3,4,1,2] => 4
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}} => [5,4,3,2,6,1] => 4
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}} => [4,3,2,6,5,1] => 4
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}} => [6,3,2,5,4,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}} => [6,4,3,5,2,1] => 4
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 4
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 5
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 5
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => {{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => 4
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => {{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => 4
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 4
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => {{1,6},{2,3,4},{5}} => [6,3,4,2,5,1] => 4
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => {{1,6},{2,3,5},{4}} => [6,3,5,4,2,1] => 4
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => {{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => {{1,5,6},{2,3,4}} => [5,3,4,2,6,1] => 4
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 5
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 4
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => {{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => 4
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => {{1,6},{2,3,4,5}} => [6,3,4,5,2,1] => 4
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 5
[] => [1] => {{1}} => [1] => 1
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
click to show known generating functions       
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see St000155The number of exceedances (also excedences) of a permutation.) plus the number of fixed points (see St000022The number of fixed points of a permutation.) of $\sigma$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.