Identifier
Values
[1] => [1,0] => [1] => [1] => 1
[1,1] => [1,0,1,0] => [2,1] => [2,1] => 2
[2] => [1,1,0,0] => [1,2] => [1,2] => 1
[1,1,1] => [1,0,1,0,1,0] => [3,2,1] => [3,2,1] => 3
[1,2] => [1,0,1,1,0,0] => [2,3,1] => [2,3,1] => 2
[2,1] => [1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 1
[3] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,2,1] => 4
[1,1,2] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,4,2,1] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => [2,4,3,1] => 2
[1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 2
[2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,4,3,2] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,1,4,2] => 2
[3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,2,4,3] => 1
[4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,5,3,2,1] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [3,5,4,2,1] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [3,4,5,2,1] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [2,5,4,3,1] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [4,2,5,3,1] => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [2,3,5,4,1] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,5,4,3,2] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [4,1,5,3,2] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [3,1,5,4,2] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [3,4,1,5,2] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [1,2,5,4,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,4,2,5,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [5,6,4,3,2,1] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [4,6,5,3,2,1] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [4,5,6,3,2,1] => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [5,3,6,4,2,1] => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [3,4,6,5,2,1] => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [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] => [2,6,5,4,3,1] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [5,2,6,4,3,1] => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [4,2,6,5,3,1] => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [4,5,2,6,3,1] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [2,3,6,5,4,1] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [2,5,3,6,4,1] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [2,3,4,6,5,1] => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => [5,1,6,4,3,2] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => [4,1,6,5,3,2] => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => [4,5,1,6,3,2] => 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => [3,1,6,5,4,2] => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [5,3,1,6,4,2] => 3
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [3,4,1,6,5,2] => 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [3,4,5,1,6,2] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => [1,5,2,6,4,3] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => [1,4,2,6,5,3] => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [1,2,5,3,6,4] => 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [6,7,5,1,2,3,4] => [1,2,6,3,7,5,4] => 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => 1
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
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Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
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.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.