Identifier
Values
[1,0] => [2,1] => [2,1] => [2,1] => 2
[1,0,1,0] => [3,1,2] => [2,3,1] => [3,2,1] => 2
[1,1,0,0] => [2,3,1] => [3,1,2] => [3,1,2] => 3
[1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 2
[1,0,1,1,0,0] => [3,1,4,2] => [4,2,3,1] => [3,4,2,1] => 4
[1,1,0,0,1,0] => [2,4,1,3] => [3,4,1,2] => [4,1,3,2] => 3
[1,1,0,1,0,0] => [4,3,1,2] => [2,4,3,1] => [3,2,4,1] => 3
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 4
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => [5,2,3,4,1] => 2
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,5,3,4,1] => [4,2,5,3,1] => 4
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,5,2,3,1] => [3,5,2,4,1] => 4
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [3,4,2,5,1] => [5,4,3,2,1] => 4
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,3,1,4] => [3,5,2,1,4] => 5
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,4,5,1,2] => [5,1,3,4,2] => 3
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,3,4,1,2] => [4,1,5,3,2] => 5
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [2,4,5,3,1] => [3,2,5,4,1] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [2,3,5,4,1] => [4,2,3,5,1] => 3
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,2,4,3,1] => [3,4,5,2,1] => 5
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,5,1,2,3] => [5,1,2,4,3] => 4
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,5,4,1,2] => [4,1,3,5,2] => 4
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [2,4,1,5,3] => [5,2,4,1,3] => 4
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 5
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 2
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,6,4,5,1] => [5,2,3,6,4,1] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,5,6,3,4,1] => [4,2,6,3,5,1] => 4
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,4,5,3,6,1] => [6,2,5,4,3,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,6,3,4,1,5] => [4,2,6,3,1,5] => 5
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,5,6,2,3,1] => [3,6,2,4,5,1] => 4
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,4,5,2,3,1] => [3,5,2,6,4,1] => 6
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [3,4,2,5,6,1] => [6,4,3,2,5,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [3,4,5,2,6,1] => [6,5,3,4,2,1] => 4
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,3,4,2,5,1] => [5,4,6,3,2,1] => 6
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [5,6,2,3,1,4] => [3,6,2,1,5,4] => 5
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,6,5,2,3,1] => [3,5,2,4,6,1] => 5
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [3,5,2,4,6,1] => [6,5,3,2,4,1] => 5
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,2,3,1,4,5] => [3,6,2,1,4,5] => 6
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,4,5,6,1,2] => [6,1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,4,5,1,2] => [5,1,3,6,4,2] => 5
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [5,6,3,4,1,2] => [4,1,6,3,5,2] => 5
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [4,5,3,6,1,2] => [6,1,5,4,3,2] => 5
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,3,4,1,2,5] => [4,1,6,3,2,5] => 6
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [2,4,5,6,3,1] => [3,2,6,4,5,1] => 3
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [2,6,4,5,3,1] => [3,2,5,6,4,1] => 5
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [2,3,5,6,4,1] => [4,2,3,6,5,1] => 3
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 3
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [2,6,3,5,4,1] => [4,2,5,6,3,1] => 5
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,6,2,4,3,1] => [3,4,6,2,5,1] => 5
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [4,5,2,6,3,1] => [3,6,5,4,2,1] => 5
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,5,2,6,4,1] => [4,6,3,5,2,1] => 5
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [6,2,4,3,1,5] => [3,4,6,2,1,5] => 6
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,5,6,1,2,3] => [6,1,2,4,5,3] => 4
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,4,5,1,2,3] => [5,1,2,6,4,3] => 6
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [3,5,6,4,1,2] => [4,1,3,6,5,2] => 4
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [3,4,6,5,1,2] => [5,1,3,4,6,2] => 4
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [6,3,5,4,1,2] => [4,1,5,6,3,2] => 6
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [2,4,1,5,6,3] => [6,2,4,1,5,3] => 4
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [2,4,5,1,6,3] => [6,2,5,4,1,3] => 4
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [2,3,6,5,4,1] => [4,2,3,5,6,1] => 4
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,2,4,1,5,3] => [5,4,6,2,1,3] => 6
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,6,1,2,3,4] => [6,1,2,3,5,4] => 5
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [4,6,5,1,2,3] => [5,1,2,4,6,3] => 5
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [3,5,1,2,6,4] => [6,1,3,5,2,4] => 5
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [2,5,1,4,6,3] => [6,2,5,1,4,3] => 5
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 6
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Description
The number of non-fixed points of a permutation.
In other words, this statistic is $n$ minus the number of fixed points (St000022The number of fixed points of a permutation.) of $\pi$.
Map
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
  • $des$ is the number of descents, St000021The number of descents of a permutation.,
  • $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
  • $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
  • $Ebot$ is the sum of the excedance bottoms,
  • $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
  • $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
  • $Res$ is the sum of the (right) embracing numbers,
  • $Ine$ is the sum of the side numbers.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.