Identifier
Values
[1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 0
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => 0
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => 0
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,3,1,5,2] => 0
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,4,3,2] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,2,1,4] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,4,3,1,6,2] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,4,2] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [4,6,3,2,1,5] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,4,3,6,5,2] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [4,1,5,3,2] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,5,1,4,2] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,5,4,1,3] => 0
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,6,3,2,5,4] => 1
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [5,1,3,6,4,2] => 1
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,4,6,1,5,2] => 0
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,4,2,6,1,3] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 0
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,6,1,2,5,3] => 0
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [5,3,6,2,1,4] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [2,6,3,5,1,4] => 1
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,3,6,5,4,2] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [4,6,3,1,5,2] => 0
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [2,4,6,5,1,3] => 0
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,1,6,2,4,3] => 0
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,6,2,5,4,3] => 0
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [3,6,5,2,1,4] => 0
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => 0
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,4,6,5,3,2] => 0
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [5,3,6,1,4,2] => 0
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,6,3,5,4,2] => 1
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [5,2,6,4,1,3] => 0
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,5,2,4,3] => 0
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => 0
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,6,1,5,3,2] => 0
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [3,6,5,1,4,2] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,6,5,4,1,3] => 0
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0
[] => [] => [1] => [1] => 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
Description
The number of non-records in a permutation.
A record in a permutation $\pi$ is a value $\pi(j)$ which is a left-to-right minimum, a left-to-right maximum, a right-to-left minimum, or a right-to-left maximum.
For example, in the permutation $\pi = [1, 4, 3, 2, 5]$, the values $1$ is a left-to-right minimum, $1, 4, 5$ are left-to-right maxima, $5, 2, 1$ are right-to-left minima and $5$ is a right-to-left maximum. Hence, $3$ is the unique non-record.
Permutations without non-records are called square [1].
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.