Identifier
Values
[1,0] => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0] => [3,1,2] => [3,1,2] => [3,1,2] => 2
[1,1,0,0] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [3,4,1,2] => [4,1,3,2] => 3
[1,0,1,1,0,0] => [3,1,4,2] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [4,1,3,2] => [3,4,1,2] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [3,4,5,1,2] => [5,1,4,3,2] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [3,4,1,2,5] => [4,1,3,2,5] => 3
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,5,2,1,4] => [5,2,3,1,4] => 4
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [3,5,1,4,2] => [4,5,1,3,2] => 3
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [4,2,5,1,3] => [2,5,1,4,3] => 3
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,1,3,5] => [2,4,1,3,5] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [4,5,3,1,2] => [3,5,1,4,2] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,5,1,3,2] => [5,1,3,4,2] => 4
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,1,3,2,5] => [3,4,1,2,5] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,1,4] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [5,2,1,4,3] => [2,4,5,1,3] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,1,3,4,2] => [3,4,5,1,2] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [3,4,5,6,1,2] => [6,1,5,4,3,2] => 5
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [3,4,5,1,2,6] => [5,1,4,3,2,6] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [3,4,6,2,1,5] => [6,2,4,3,1,5] => 5
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [3,4,6,1,5,2] => [5,6,1,4,3,2] => 4
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [3,4,1,2,5,6] => [4,1,3,2,5,6] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,5,2,6,1,4] => [6,2,3,1,5,4] => 5
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,5,2,1,4,6] => [5,2,3,1,4,6] => 4
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [3,5,6,4,1,2] => [4,6,1,5,3,2] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [3,5,6,1,4,2] => [6,1,4,5,3,2] => 5
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [3,5,1,4,2,6] => [4,5,1,3,2,6] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,6,2,4,1,5] => [4,6,2,3,1,5] => 4
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,6,2,1,5,4] => [5,6,2,3,1,4] => 4
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [3,6,1,4,5,2] => [4,5,6,1,3,2] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [4,2,5,6,1,3] => [2,6,1,5,4,3] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [4,2,5,1,3,6] => [2,5,1,4,3,6] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,6,3,1,5] => [2,6,3,4,1,5] => 4
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [4,2,6,1,5,3] => [2,5,6,1,4,3] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,1,3,5,6] => [2,4,1,3,5,6] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [4,5,3,6,1,2] => [3,6,1,5,4,2] => 4
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [4,5,3,1,2,6] => [3,5,1,4,2,6] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,5,6,3,1,2] => [6,3,5,1,4,2] => 5
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [4,5,6,1,2,3] => [6,1,5,2,4,3] => 5
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,5,1,3,2,6] => [5,1,3,4,2,6] => 4
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,6,3,2,1,5] => [3,6,2,4,1,5] => 4
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [4,6,3,1,5,2] => [3,5,6,1,4,2] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,1,3,5,2] => [5,6,1,3,4,2] => 4
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,1,3,2,5,6] => [3,4,1,2,5,6] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [5,2,3,6,1,4] => [2,3,6,1,5,4] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,1,4,6] => [2,3,5,1,4,6] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [5,2,6,4,1,3] => [2,4,6,1,5,3] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,6,1,4,3] => [2,6,1,4,5,3] => 4
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [5,2,1,4,3,6] => [2,4,5,1,3,6] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [5,6,3,4,1,2] => [3,4,6,1,5,2] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,6,3,1,4,2] => [3,6,1,4,5,2] => 4
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,6,1,4,3,2] => [4,6,1,3,5,2] => 4
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,1,3,4,2,6] => [3,4,5,1,2,6] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => [2,3,4,6,1,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [6,2,3,1,5,4] => [2,3,5,6,1,4] => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [6,2,1,4,5,3] => [2,4,5,6,1,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,1,3,4,5,2] => [3,4,5,6,1,2] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => [4,1,3,2,5,6,7] => [3,4,1,2,5,6,7] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => [2,3,4,5,7,1,6] => 2
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [7,2,3,4,1,6,5] => [2,3,4,6,7,1,5] => 2
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [7,2,3,1,5,6,4] => [2,3,5,6,7,1,4] => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [7,1,3,4,5,6,2] => [3,4,5,6,7,1,2] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [8,1,2,3,7,4,5,6] => [3,4,5,7,8,1,6,2] => [8,1,6,7,5,4,3,2] => 7
[1,0,1,1,1,0,0,1,1,0,1,0,0,0] => [3,1,8,6,2,7,4,5] => [3,6,2,8,1,5,7,4] => [7,8,2,3,1,5,6,4] => 6
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [2,7,1,8,3,4,5,6] => [4,2,6,7,8,1,3,5] => [2,8,1,7,4,3,6,5] => 6
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [8,3,1,2,4,5,6,7] => [4,5,3,6,7,8,1,2] => [3,8,1,7,6,5,4,2] => 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [4,3,1,5,6,7,8,2] => [4,1,3,2,5,6,7,8] => [3,4,1,2,5,6,7,8] => 2
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [7,8,4,1,6,2,3,5] => [5,7,8,4,1,6,2,3] => [4,6,8,1,7,2,5,3] => 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [7,8,5,1,6,2,3,4] => [5,7,8,1,4,6,2,3] => [6,8,1,4,7,2,5,3] => 6
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => [8,2,3,4,5,6,1,7] => [2,3,4,5,6,8,1,7] => 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [2,3,4,5,8,7,1,6] => [8,2,3,4,5,1,7,6] => [2,3,4,5,7,8,1,6] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [8,3,4,5,6,7,1,2] => [8,1,3,4,5,6,7,2] => [3,4,5,6,7,8,1,2] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [4,3,1,5,6,7,8,9,2] => [4,1,3,2,5,6,7,8,9] => [3,4,1,2,5,6,7,8,9] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => [9,2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,9,1,8] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[] => [1] => [1] => [1] => 0
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [4,3,1,5,6,7,8,9,10,2] => [4,1,3,2,5,6,7,8,9,10] => [3,4,1,2,5,6,7,8,9,10] => 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
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Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
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].
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.