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Identifier
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,0] => [2,3,1] => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0] => [3,1,2] => [2,3,1] => [2,3,1] => 1
[1,1,0,1,0,0] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [3,4,1,2] => [3,4,1,2] => 1
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [4,2,3,1] => [4,2,3,1] => 2
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let π be a permutation. Following Viennot [1], we associate to π a heap of pieces, by considering each decreasing run (πi,πi+1,…,πj) of π 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.