Identifier
Values
[1] => [1,0] => [2,1] => 1
[2] => [1,0,1,0] => [3,1,2] => 1
[1,1] => [1,1,0,0] => [2,3,1] => 1
[3] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 4
[1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 4
[4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 4
[2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 4
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 5
[5] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 4
[3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 5
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 4
[2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 6
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 5
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 1
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => 4
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 5
[3,3] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 4
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 11
[2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 8
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => 7
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => 1
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => 4
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 5
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 4
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 5
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 6
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 6
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => 4
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => 8
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => 1
[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [8,1,2,3,4,5,6,9,7] => 4
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 10
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 8
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 4
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,9,1,2,3,4,5,6,7] => 1
[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => 1
[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,1,2,3,4,5,6,7,10,8] => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,9,1,2,3,4,5,6,7,8] => 10
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => 1
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [8,1,2,7,6,3,4,5] => 10
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => 11
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [6,7,8,1,2,3,4,5] => 1
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[7,7] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [9,7,8,1,2,3,4,5,6] => 8
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [8,7,4,5,6,1,2,3] => 12
[8,8] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [10,9,8,1,2,3,4,5,6,7] => 18
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 1
[2,2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 1
[] => [] => [1] => 0
[4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[5,5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => 1
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Description
Babson and Steingrímsson's statistic of a permutation.
In terms of generalized patterns this is
$$ (13-2) + (21-3) + (32-1) + (21). $$
Here, $(\pi)$ denotes the number of times the pattern $\pi$ occurs in a permutation, and letters in the pattern which are not separated by a dash must appear consecutively.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.