Identifier
Values
[1,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [2,3,1] => [3,1,2] => 2
[1,1,0,0] => [2,3,1] => [3,1,2] => [2,3,1] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => [3,4,1,2] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => [4,2,3,1] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => [4,2,1,3] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,3,5,1,4] => [4,5,1,2,3] => 3
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [2,4,1,5,3] => [5,3,4,1,2] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [2,4,5,3,1] => [5,3,1,2,4] => 3
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,5,1,3,4] => [3,4,5,1,2] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,4,5,2] => [5,2,3,1,4] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,1,5,2,4] => [4,5,2,3,1] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,4,2,5,1] => [5,1,4,2,3] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,4,5,2,1] => [5,2,1,3,4] => 3
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,1,4] => [4,5,2,1,3] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,1,2,5,3] => [2,5,3,4,1] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,1,5,3,2] => [5,3,2,4,1] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,5,2,3,1] => [3,1,5,2,4] => 3
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,4,6,1,5] => [5,6,1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,3,5,1,6,4] => [6,4,5,1,2,3] => 3
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,3,5,6,4,1] => [6,4,1,2,3,5] => 4
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,3,6,1,4,5] => [4,5,6,1,2,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [2,4,1,5,6,3] => [6,3,4,1,2,5] => 3
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [2,4,1,6,3,5] => [5,6,3,4,1,2] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [2,4,5,3,6,1] => [6,1,2,5,3,4] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [2,4,5,6,3,1] => [6,3,1,2,4,5] => 4
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [2,4,6,3,1,5] => [5,6,3,1,2,4] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [2,5,1,3,6,4] => [3,6,4,5,1,2] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [2,5,1,6,4,3] => [6,4,3,5,1,2] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [2,5,6,3,4,1] => [4,1,2,6,3,5] => 4
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [2,6,1,3,4,5] => [3,4,5,6,1,2] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => [6,2,3,1,4,5] => 3
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,1,4,6,2,5] => [5,6,2,3,1,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,1,5,2,6,4] => [6,4,5,2,3,1] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,1,5,6,4,2] => [6,4,2,3,1,5] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,1,6,2,4,5] => [4,5,6,2,3,1] => 3
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => [6,1,4,2,3,5] => 4
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,4,2,6,1,5] => [5,6,1,4,2,3] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => [6,1,5,2,3,4] => 4
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [3,4,5,6,1,2] => [2,6,1,3,4,5] => 4
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [3,4,6,2,1,5] => [5,6,2,1,3,4] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,5,2,1,6,4] => [6,4,5,2,1,3] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => [6,4,1,5,2,3] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,5,6,2,4,1] => [4,1,6,2,3,5] => 4
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,6,2,1,4,5] => [4,5,6,2,1,3] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,1,2,5,6,3] => [2,6,3,4,1,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,1,2,6,3,5] => [2,5,6,3,4,1] => 3
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,1,5,3,6,2] => [6,2,5,3,4,1] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,1,5,6,3,2] => [6,3,2,4,1,5] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,1,6,3,2,5] => [5,6,3,2,4,1] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,5,2,3,6,1] => [6,1,3,5,2,4] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [4,5,2,6,3,1] => [6,3,1,5,2,4] => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => [6,3,2,1,4,5] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,6,2,3,1,5] => [3,1,5,6,2,4] => 3
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,1,2,3,6,4] => [2,3,6,4,5,1] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,1,2,6,4,3] => [2,6,4,3,5,1] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,1,6,3,4,2] => [4,2,6,3,5,1] => 3
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,6,2,3,4,1] => [4,1,3,6,2,5] => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [2,3,4,6,7,5,1] => [7,5,1,2,3,4,6] => 5
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [2,3,5,6,7,4,1] => [7,4,1,2,3,5,6] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [2,5,6,7,4,3,1] => [7,4,3,1,2,5,6] => 4
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => [7,2,3,1,4,5,6] => 4
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [3,1,4,6,7,5,2] => [7,5,2,3,1,4,6] => 3
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [3,1,5,6,7,4,2] => [7,4,2,3,1,5,6] => 3
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [7,2,1,3,4,5,6] => 5
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => 6
[] => [1] => [1] => [1] => 1
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Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also St000325The width of the tree associated to a permutation. for the width of this tree.
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
inverse
Description
Sends a permutation to its inverse.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.