Identifier
Values
[1] => [1,0] => [1,0] => [1] => 0
[1,1] => [1,0,1,0] => [1,1,0,0] => [1,2] => 0
[2] => [1,1,0,0] => [1,0,1,0] => [2,1] => 0
[1,1,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,2] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => [2,1,3] => 0
[2,1] => [1,1,0,0,1,0] => [1,1,0,1,0,0] => [3,1,2] => 0
[3] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [2,3,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => 0
[1,3] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => 1
[4] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 0
[1,1,1,1,1] => [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] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 2
[5] => [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] => 0
[1,1,1,1,1,1] => [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] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [3,1,2,4,5,6] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,1,2,5,6] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4,6] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [3,4,5,1,2,6] => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,6,1,3,4,5] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [3,6,1,2,4,5] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [3,4,6,1,2,5] => 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [3,5,6,1,2,4] => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => 3
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => 3
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 0
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 restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.