Identifier
Values
[1,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,0] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [4,1,3,2] => [2,4,3,1] => 1
[1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => [4,3,1,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,1,2,4,3] => [2,3,5,4,1] => 1
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,1,3,2,4] => [2,4,3,5,1] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,1,5,3,2] => [2,5,4,1,3] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,1,3,4,2] => [2,5,3,4,1] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => [3,2,4,5,1] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,2,1,4,3] => [3,2,5,4,1] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,2,1,4] => [4,3,1,5,2] => 2
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,5,2,3,1] => [5,3,4,1,2] => 2
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,4,1] => [5,3,1,4,2] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4] => [4,2,3,5,1] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,5,3,1] => [5,2,4,1,3] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,5,4,2,1] => [5,4,1,3,2] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => 2
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,1,2,3,5,4] => [2,3,4,6,5,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,1,2,4,3,5] => [2,3,5,4,6,1] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,1,2,6,4,3] => [2,3,6,5,1,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,1,2,4,5,3] => [2,3,6,4,5,1] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,1,3,2,4,5] => [2,4,3,5,6,1] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,1,3,2,5,4] => [2,4,3,6,5,1] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,1,6,3,2,5] => [2,5,4,1,6,3] => 2
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,1,6,3,4,2] => [2,6,4,5,1,3] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [4,1,6,3,5,2] => [2,6,4,1,5,3] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,1,3,4,2,5] => [2,5,3,4,6,1] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,1,3,6,4,2] => [2,6,3,5,1,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [4,1,6,5,3,2] => [2,6,5,1,4,3] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,1,3,4,5,2] => [2,6,3,4,5,1] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => [3,2,4,5,6,1] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,2,1,3,5,4] => [3,2,4,6,5,1] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,2,1,4,3,5] => [3,2,5,4,6,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,2,1,6,4,3] => [3,2,6,5,1,4] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,2,1,4,5,3] => [3,2,6,4,5,1] => 3
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,6,2,1,4,5] => [4,3,1,5,6,2] => 2
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,6,2,1,5,4] => [4,3,1,6,5,2] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,6,2,3,1,5] => [5,3,4,1,6,2] => 3
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,5,2,3,4,1] => [6,3,4,5,2,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,6,2,3,5,1] => [6,3,4,1,5,2] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,6,2,4,1,5] => [5,3,1,4,6,2] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => [6,3,1,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,2,5,3,1] => [6,3,5,1,4,2] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,6,2,4,5,1] => [6,3,1,4,5,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,2,3,1,4,5] => [4,2,3,5,6,1] => 2
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,2,3,1,5,4] => [4,2,3,6,5,1] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,6,3,1,5] => [5,2,4,1,6,3] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,6,3,4,1] => [6,2,4,5,1,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,6,3,5,1] => [6,2,4,1,5,3] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,6,5,2,4,1] => [6,4,1,5,3,2] => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [3,6,4,2,5,1] => [6,4,1,3,5,2] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => [5,2,3,4,6,1] => 2
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,3,6,4,1] => [6,2,3,5,1,4] => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [4,2,6,5,3,1] => [6,2,5,1,4,3] => 3
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,6,4,5,2,1] => [6,5,1,3,4,2] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => [4,2,3,5,6,7,1] => 2
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => [5,2,3,4,6,7,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => [6,2,3,4,5,7,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [5,1,2,3,6,7,8,4] => [8,1,2,3,5,6,7,4] => [2,3,4,8,5,6,7,1] => 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [4,1,2,5,6,8,3,7] => [8,1,2,4,5,6,3,7] => [2,3,7,4,5,6,8,1] => 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [7,1,4,2,6,3,8,5] => [4,1,6,3,8,5,7,2] => [2,8,4,1,6,3,7,5] => 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0] => [8,1,6,5,2,3,4,7] => [5,1,6,8,4,3,2,7] => [2,7,6,5,1,3,8,4] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => [8,2,1,3,4,5,6,7] => [3,2,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,8,5,7] => [8,2,1,4,3,6,5,7] => [3,2,5,4,7,6,8,1] => 4
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [8,4,1,2,7,3,5,6] => [4,7,2,3,8,5,6,1] => [8,3,4,1,6,7,2,5] => 3
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [8,7,1,5,2,3,4,6] => [5,7,2,8,4,3,6,1] => [8,3,6,5,1,7,2,4] => 3
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [6,8,1,5,2,7,3,4] => [6,5,2,8,4,7,3,1] => [8,3,7,5,2,1,6,4] => 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [2,3,8,1,4,5,6,7] => [8,2,3,1,4,5,6,7] => [4,2,3,5,6,7,8,1] => 2
[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [2,6,8,1,3,7,4,5] => [8,2,6,3,4,7,5,1] => [8,2,4,5,7,3,6,1] => 3
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [8,3,4,1,6,7,2,5] => [3,6,4,2,8,7,5,1] => [8,4,1,3,7,2,6,5] => 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [2,3,4,8,1,5,6,7] => [8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => 2
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [2,7,6,5,1,3,8,4] => [5,2,6,8,4,3,7,1] => [8,2,6,5,1,3,7,4] => 3
[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [2,8,4,5,7,1,3,6] => [4,2,8,5,7,3,6,1] => [8,2,6,1,4,7,5,3] => 3
[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] => [9,1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,9,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => [9,2,1,3,4,5,6,7,8] => [3,2,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,9,1,4,5,6,7,8] => [9,2,3,1,4,5,6,7,8] => [4,2,3,5,6,7,8,9,1] => 2
[] => [1] => [1] => [1] => 0
[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] => [10,1,2,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,9,10,1] => 1
[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] => [11,1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => [10,2,1,3,4,5,6,7,8,9] => [3,2,4,5,6,7,8,9,10,1] => 2
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Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with St000366The number of double descents of a permutation. is studied in [3], where left outer peaks are called exterior peaks.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
Map
inverse
Description
Sends a permutation to its inverse.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.