Identifier
Values
[1,0] => [2,1] => [2,1] => 0
[1,0,1,0] => [3,1,2] => [3,1,2] => 0
[1,1,0,0] => [2,3,1] => [3,2,1] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [4,3,1,2] => 0
[1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 0
[1,1,0,1,0,0] => [4,3,1,2] => [3,1,4,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,4,1,2,3] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,3,1,2,4] => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,1,2,5,3] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,5,2,4] => 0
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,5,2,3] => 0
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,1,5,4,2] => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,3,2,1,4] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,1,5,3] => 0
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,3,1,5,2] => 0
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,4,1,2,3,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,1,2,3,6,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,5,4,1,2,3] => 0
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,3,1,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,5,3,1,2,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,1,2,6,3,5] => 0
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,1,2,6,3,4] => 0
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [4,1,2,6,5,3] => 0
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,4,3,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,3,1,2,6,4] => 0
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [5,4,1,2,6,3] => 0
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,5,4,3,1,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,5,2,1,3,4] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,4,2,1,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,2,1,3,6,4] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,5,4,2,1,3] => 0
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 0
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,1,6,5,2,4] => 0
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,6,2,3,5] => 0
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,1,5,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,1,6,5,2,3] => 0
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,1,6,4,2,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,1,5,2,6,4] => 0
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,4,1,6,2,3] => 0
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,1,6,5,4,2] => 0
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,3,2,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,5,3,2,1,4] => 1
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,1,6,3,5] => 0
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,1,6,3,4] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,1,6,5,3] => 0
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,3,1,6,2,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,3,1,6,2,4] => 0
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,1,5,2,6,3] => 0
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,3,1,6,5,2] => 0
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,4,3,2,1,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,3,2,1,6,4] => 0
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,4,2,1,6,3] => 0
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,4,3,1,6,2] => 0
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,5,4,3,2,1] => 0
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Description
The number of 3-rises of a permutation.
A 3-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+3 = \pi(i+1)$.
For 1-rises, or successions, see St000441The number of successions of a permutation., for 2-rises see St000534The number of 2-rises of a permutation..
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].