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] => [2,1,3] => 0
[1,2] => [1,0,1,1,0,0] => [2,3,1] => 1
[2,1] => [1,1,0,0,1,0] => [3,1,2] => 1
[3] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => 1
[1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => 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] => [3,1,4,2] => 1
[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] => 1
[4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => 1
[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] => [4,1,5,2,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,5,6] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,4,2,6,5] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [3,1,4,6,2,5] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,4,6,1,2,5] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [3,4,1,2,5,6] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,4,5,6,2] => 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [4,1,5,2,6,3] => 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,5,1,2,6,3] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [4,1,5,6,2,3] => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,1,6,2,3,4] => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 0
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Description
The number of cycles of length at least 3 of a permutation.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.