Identifier
Values
[1,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [3,1,2] => 1
[1,1,0,0] => [2,3,1] => [1,3,2] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [2,1,4,3] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,0] => [4,3,1,2] => [4,3,1,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [1,2,4,3] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [3,1,2,5,4] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,1,3,5,4] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,3,2,5,4] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,3,1,2,4] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,2,1,5,4] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,3,5,2,4] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,5,4,1,3] => 1
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,2,4,1,3] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [4,1,2,3,6,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,2,6,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [3,1,2,4,6,5] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,6,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [6,1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [4,1,3,2,6,5] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [2,1,4,6,3,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,1,6,5,2,4] => 1
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,1,3,5,2,4] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [2,1,3,4,6,5] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,6,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,4,1,3,6,5] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,4,1,6,3,5] => 1
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,1,5,3,4] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,3,2,4,6,5] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [4,3,1,2,6,5] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [6,4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [4,6,1,2,3,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,3,1,2,6,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,6,2,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [6,3,1,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [6,3,1,5,2,4] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,2,1,4,6,5] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,3,6,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,2,4,3,6,5] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,6,4,1,3,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,6,5,1,3,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,4,3,2,6,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,2,4,1,3,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [6,3,5,1,2,4] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [6,5,4,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,1,3,2,6,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,2,4,6,3,5] => 1
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,3,6,5,2,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,6,3,5,2,4] => 1
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,1,3,5,2,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1
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Description
The number of factors of the Stanley symmetric function associated with a permutation.
For example, the Stanley symmetric function of $\pi=321645$ equals
$20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$
Map
Inverse fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.