Identifier
Values
[1,0] => [1] => [1] => [-1] => 0
[1,0,1,0] => [2,1] => [2,1] => [1,-2] => 1
[1,1,0,0] => [1,2] => [1,2] => [2,-1] => 1
[1,0,1,0,1,0] => [2,1,3] => [2,1,3] => [1,3,-2] => 2
[1,0,1,1,0,0] => [2,3,1] => [2,3,1] => [1,2,-3] => 2
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => [3,1,-2] => 2
[1,1,0,1,0,0] => [1,3,2] => [1,3,2] => [3,2,-1] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [2,3,-1] => 2
[1,0,1,0,1,0,1,0] => [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 2
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 3
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 3
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 3
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 3
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => 2
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 3
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => 3
[1,1,0,1,0,1,0,0] => [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 2
[1,1,0,1,1,0,0,0] => [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 2
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => 3
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => 2
[1,1,1,0,1,0,0,0] => [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 3
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Description
The number of minimal elements in Bruhat order not less than the signed permutation.
The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent.
This is the size of the essential set of the signed permutation, see [1].
Map
to signed permutation
Description
The signed permutation with all signs positive.
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
inverse Kreweras complement
Description
The inverse Kreweras complement of a signed permutation.
This is the signed permutation $c \pi^{-1}$ where $c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the inverse Kreweras complement on signed permutations of $\{\pm 1,\dots, \pm n\}$ is $2n$.