Identifier
Values
[1,1] => [1,0,1,0] => [2,1] => 0
[2] => [1,1,0,0] => [1,2] => 0
[1,1,1] => [1,0,1,0,1,0] => [3,2,1] => 0
[1,2] => [1,0,1,1,0,0] => [2,3,1] => 0
[2,1] => [1,1,0,0,1,0] => [3,1,2] => 0
[3] => [1,1,1,0,0,0] => [1,2,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 0
[1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 0
[4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 6
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 6
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 3
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 10
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 15
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Description
The number of occurrences of the vincular pattern |123 in a permutation.
This is the number of occurrences of the pattern $(1,2,3)$, such that the letter matched by $1$ is the first entry of the permutation.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
Map
bounce path
Description
The bounce path determined by an integer composition.