Identifier
-
Mp00007:
Alternating sign matrices
—to Dyck path⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001860: Signed permutations ⟶ ℤ
Values
[[1,0],[0,1]] => [1,0,1,0] => [2,1] => [2,1] => 1
[[0,1],[1,0]] => [1,1,0,0] => [1,2] => [1,2] => 0
[[1,0,0],[0,1,0],[0,0,1]] => [1,0,1,0,1,0] => [3,2,1] => [3,2,1] => 1
[[0,1,0],[1,0,0],[0,0,1]] => [1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 2
[[1,0,0],[0,0,1],[0,1,0]] => [1,0,1,1,0,0] => [2,3,1] => [2,3,1] => 2
[[0,1,0],[1,-1,1],[0,1,0]] => [1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 1
[[0,0,1],[1,0,0],[0,1,0]] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[[0,1,0],[0,0,1],[1,0,0]] => [1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 1
[[0,0,1],[0,1,0],[1,0,0]] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,2,1] => 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => [4,3,1,2] => 3
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => [4,2,3,1] => 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => [4,2,1,3] => 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => [4,2,1,3] => 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,4,2,1] => 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,4,1,2] => 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => [2,3,1,4] => 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => [2,3,1,4] => 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => 4
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => [2,3,1,4] => 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => [2,3,1,4] => 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => [2,3,1,4] => 2
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
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Description
The number of factors of the Stanley symmetric function associated with a signed permutation.
Map
to Dyck path
Description
The Dyck path determined by the last diagonal of the monotone triangle of an alternating sign matrix.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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