Identifier
Values
[1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 1
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 1
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [1,3,4,2] => 1
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4] => 1
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,3,4,5,2] => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,2,5,4] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,2,4,3] => 1
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,1,4,2,6,5] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,1,5,3,4] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,5,2,4] => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,2,4,5,3] => 1
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,1,4,2,5,3] => 2
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => 1
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [6,2,1,4,3,5] => 1
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,6,2,5] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => 1
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [6,3,1,5,2,4] => 2
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,5,6,1,3] => 1
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,5,2,4,6,3] => 2
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,6,3,1,4,5] => 1
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,4,3,6,5] => 1
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,1,3,4,2,5] => 1
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,3,2,5,6,4] => 1
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [6,2,1,5,3,4] => 1
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,2,6,3,5,4] => 1
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,2,4,5,6,3] => 1
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [1,2,5,4,3,7,6] => 1
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,3,2,4,6,5] => 1
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,6,2,1,3,5] => 1
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,4,2,3,6,5] => 1
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,1,5,6,3,4] => 1
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,5,6,2,4] => 1
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,2,3,5,4] => 1
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,1,3,6,4,5] => 1
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,6,3,2,4,5] => 1
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,2,6,3,5] => 1
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,2,3,5,6,4] => 1
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [1,2,4,5,6,7,3] => 1
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => [1,3,2,5,6,7,4] => 1
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [1,2,6,3,5,7,4] => 1
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [1,2,3,7,5,4,6] => 1
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => [1,2,4,3,6,7,5] => 1
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [1,2,6,3,4,7,5] => 1
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [1,2,4,3,5,7,6] => 1
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => [1,3,5,2,7,4,6] => 2
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,2,5,3,4,7,6] => 1
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => [1,3,2,4,6,7,5] => 1
[4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [1,4,2,3,6,7,5] => 1
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,2,7,3,4,6,5] => 1
[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [1,2,3,5,6,7,4] => 1
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [1,3,2,4,5,7,6] => 1
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,4,2,3,5,7,6] => 1
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,3,2,7,4,5,6] => 1
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [1,4,2,7,3,5,6] => 1
[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [1,2,3,4,6,7,5] => 1
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 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
Description
The neighbouring number of a permutation.
For a permutation $\pi$, this is
$$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
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.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.