Identifier
Values
[1,0] => [1] => [1] => [1] => 1
[1,0,1,0] => [2,1] => [2,1] => [-2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => [1,2] => 2
[1,0,1,0,1,0] => [2,3,1] => [2,3,1] => [-3,-2,1] => 1
[1,0,1,1,0,0] => [2,1,3] => [2,1,3] => [-2,1,3] => 2
[1,1,0,0,1,0] => [1,3,2] => [1,3,2] => [-3,1,2] => 2
[1,1,0,1,0,0] => [3,1,2] => [3,1,2] => [3,1,2] => 3
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [2,3,4,1] => [-4,-3,-2,1] => 1
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [2,3,1,4] => [-3,-2,1,4] => 2
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => 2
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [2,4,1,3] => [2,-4,1,3] => 3
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => 3
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => 2
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => 3
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [3,1,4,2] => [4,-3,1,2] => 3
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 4
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => 3
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => [4,1,2,3] => 4
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [4,1,2,3] => [1,-4,2,3] => 3
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [2,3,4,5,1] => [-5,-4,-3,-2,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [2,3,4,1,5] => [-4,-3,-2,1,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [2,3,1,5,4] => [-5,-3,-2,1,4] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [2,3,5,1,4] => [3,-5,-2,1,4] => 3
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [2,3,1,4,5] => [-3,-2,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,4,5,3] => [-5,-4,-2,1,3] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => [-4,-2,1,3,5] => 3
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [2,4,1,5,3] => [2,-5,-4,1,3] => 3
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [2,4,5,1,3] => [2,5,-4,1,3] => 4
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [2,4,1,3,5] => [2,-4,1,3,5] => 4
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => [-5,-2,1,3,4] => 3
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,5,3,4] => [2,-5,1,3,4] => 4
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,1,3,4] => 3
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,1,3,4,5] => 4
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,3,4,5,2] => [-5,-4,-3,1,2] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,3,4,2,5] => [-4,-3,1,2,5] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => [-5,-3,1,2,4] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,3,5,2,4] => [3,-5,1,2,4] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => [-3,1,2,4,5] => 4
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [3,1,4,5,2] => [5,-4,-3,1,2] => 3
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [3,1,4,2,5] => [4,-3,1,2,5] => 4
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [3,4,1,5,2] => [4,5,-3,1,2] => 4
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => 5
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => 5
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [3,1,2,5,4] => [5,-3,1,2,4] => 4
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [3,1,5,2,4] => [3,5,1,2,4] => 5
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [3,5,1,2,4] => [-5,3,1,2,4] => 4
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 5
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,4,5,3] => [-5,-4,1,2,3] => 3
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => [-4,1,2,3,5] => 4
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,4,2,5,3] => [5,-4,1,2,3] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => 5
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => [4,1,2,3,5] => 5
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [4,1,2,5,3] => [1,-5,-4,2,3] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [4,1,5,2,3] => [-4,5,1,2,3] => 4
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => [-5,1,-4,2,3] => 3
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [4,1,2,3,5] => [1,-4,2,3,5] => 4
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => [5,1,2,3,4] => 5
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => [1,-5,2,3,4] => 4
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [5,1,2,3,4] => [1,5,2,3,4] => 5
[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] => 5
[1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => 6
[1,1,0,1,0,1,0,1,1,0,0,0] => [3,4,5,1,2,6] => [3,4,5,1,2,6] => [3,4,5,1,2,6] => 6
[1,1,0,1,0,1,1,0,0,1,0,0] => [3,4,1,6,2,5] => [3,4,1,6,2,5] => [3,4,6,1,2,5] => 6
[1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,4,1,2,5,6] => 6
[1,1,0,1,1,0,0,1,0,1,0,0] => [3,1,5,6,2,4] => [3,1,5,6,2,4] => [3,5,6,1,2,4] => 6
[1,1,0,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4,6] => [3,1,5,2,4,6] => [3,5,1,2,4,6] => 6
[1,1,0,1,1,1,0,0,0,1,0,0] => [3,1,2,6,4,5] => [3,1,2,6,4,5] => [3,6,1,2,4,5] => 6
[1,1,0,1,1,1,0,1,0,0,0,0] => [3,6,1,2,4,5] => [3,6,1,2,4,5] => [6,3,1,2,4,5] => 6
[1,1,0,1,1,1,1,0,0,0,0,0] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => 6
[1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,5,6,1,2,3] => 6
[1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,5,1,2,3,6] => 6
[1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,6,3,5] => [1,4,2,6,3,5] => [4,6,1,2,3,5] => 6
[1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [4,1,2,3,5,6] => 6
[1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,6,1,2,3,4] => 6
[1,1,1,1,0,0,0,1,1,0,0,0] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => [5,1,2,3,4,6] => 6
[1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => [5,1,2,6,3,4] => [1,5,6,2,3,4] => 6
[1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => [5,6,1,2,3,4] => [5,1,6,2,3,4] => 6
[1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => [1,5,2,3,4,6] => 6
[1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [6,1,2,3,4,5] => 6
[1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 6
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [3,4,7,1,2,5,6] => [3,4,7,1,2,5,6] => [7,3,4,1,2,5,6] => 7
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [3,6,1,2,7,4,5] => [3,6,1,2,7,4,5] => [6,3,7,1,2,4,5] => 7
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,7,2,4,5,6] => [3,1,7,2,4,5,6] => [7,3,1,2,4,5,6] => 7
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [1,4,7,2,3,5,6] => [1,4,7,2,3,5,6] => [7,4,1,2,3,5,6] => 7
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [5,7,1,2,3,4,6] => 7
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [1,6,7,2,3,4,5] => [1,6,7,2,3,4,5] => [6,1,7,2,3,4,5] => 7
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [7,1,2,3,4,5,6] => 7
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0] => [1,2,5,6,7,8,3,4] => [1,2,5,6,7,8,3,4] => [5,6,7,8,1,2,3,4] => 8
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Description
The number of positive entries in a signed permutation.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
Foata-Han inverse
Description
The inverse of the Foata-Han bijection for signed permutations.
This map sends the the flag inversion number St001428The number of B-inversions of a signed permutation. to the flag major index St001433The flag major index of a signed permutation.