Identifier
-
Mp00161:
Signed permutations
—reverse⟶
Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[3,2,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[3,2,-1] => [-1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[3,-2,1] => [1,-2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[3,-2,-1] => [-1,-2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[-3,2,1] => [1,2,-3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[-3,2,-1] => [-1,2,-3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[-3,-2,1] => [1,-2,-3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[-3,-2,-1] => [-1,-2,-3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,2,3,-1] => [-1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,2,-3,1] => [1,-3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,2,-3,-1] => [-1,-3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,-2,3,1] => [1,3,-2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,-2,3,-1] => [-1,3,-2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,-2,-3,1] => [1,-3,-2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,-2,-3,-1] => [-1,-3,-2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[-4,2,3,1] => [1,3,2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[-4,2,3,-1] => [-1,3,2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[-4,2,-3,1] => [1,-3,2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[-4,2,-3,-1] => [-1,-3,2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[-4,-2,3,1] => [1,3,-2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[-4,-2,3,-1] => [-1,3,-2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[-4,-2,-3,1] => [1,-3,-2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[-4,-2,-3,-1] => [-1,-3,-2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[4,3,2,-1] => [-1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[4,3,-2,1] => [1,-2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[4,3,-2,-1] => [-1,-2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[4,-3,2,1] => [1,2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[4,-3,2,-1] => [-1,2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[4,-3,-2,1] => [1,-2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[4,-3,-2,-1] => [-1,-2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[-4,3,2,1] => [1,2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[-4,3,2,-1] => [-1,2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[-4,3,-2,1] => [1,-2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[-4,3,-2,-1] => [-1,-2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[-4,-3,2,1] => [1,2,-3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[-4,-3,2,-1] => [-1,2,-3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[-4,-3,-2,1] => [1,-2,-3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[-4,-3,-2,-1] => [-1,-2,-3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,2,3,4,-1] => [-1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,2,3,-4,1] => [1,-4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,2,3,-4,-1] => [-1,-4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,2,-3,4,1] => [1,4,-3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,2,-3,4,-1] => [-1,4,-3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,2,-3,-4,1] => [1,-4,-3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,2,-3,-4,-1] => [-1,-4,-3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,-2,3,4,1] => [1,4,3,-2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,-2,3,4,-1] => [-1,4,3,-2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,-2,3,-4,1] => [1,-4,3,-2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,-2,3,-4,-1] => [-1,-4,3,-2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,-2,-3,4,1] => [1,4,-3,-2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,-2,-3,4,-1] => [-1,4,-3,-2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,-2,-3,-4,1] => [1,-4,-3,-2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,-2,-3,-4,-1] => [-1,-4,-3,-2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,2,3,4,1] => [1,4,3,2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,2,3,4,-1] => [-1,4,3,2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,2,3,-4,1] => [1,-4,3,2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,2,3,-4,-1] => [-1,-4,3,2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,2,-3,4,1] => [1,4,-3,2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,2,-3,4,-1] => [-1,4,-3,2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,2,-3,-4,1] => [1,-4,-3,2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,2,-3,-4,-1] => [-1,-4,-3,2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,-2,3,4,1] => [1,4,3,-2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,-2,3,4,-1] => [-1,4,3,-2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,-2,3,-4,1] => [1,-4,3,-2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,-2,3,-4,-1] => [-1,-4,3,-2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,-2,-3,4,1] => [1,4,-3,-2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,-2,-3,4,-1] => [-1,4,-3,-2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,-2,-3,-4,1] => [1,-4,-3,-2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[-5,-2,-3,-4,-1] => [-1,-4,-3,-2,-5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[5,2,4,3,1] => [1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,2,4,3,-1] => [-1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,2,4,-3,1] => [1,-3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,2,4,-3,-1] => [-1,-3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,2,-4,3,1] => [1,3,-4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,2,-4,3,-1] => [-1,3,-4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,2,-4,-3,1] => [1,-3,-4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,2,-4,-3,-1] => [-1,-3,-4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,-2,4,3,1] => [1,3,4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,-2,4,3,-1] => [-1,3,4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,-2,4,-3,1] => [1,-3,4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,-2,4,-3,-1] => [-1,-3,4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,-2,-4,3,1] => [1,3,-4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,-2,-4,3,-1] => [-1,3,-4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,-2,-4,-3,1] => [1,-3,-4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[5,-2,-4,-3,-1] => [-1,-3,-4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,2,4,3,1] => [1,3,4,2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,2,4,3,-1] => [-1,3,4,2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,2,4,-3,1] => [1,-3,4,2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,2,4,-3,-1] => [-1,-3,4,2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,2,-4,3,1] => [1,3,-4,2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,2,-4,3,-1] => [-1,3,-4,2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,2,-4,-3,1] => [1,-3,-4,2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,2,-4,-3,-1] => [-1,-3,-4,2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,-2,4,3,1] => [1,3,4,-2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,-2,4,3,-1] => [-1,3,4,-2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,-2,4,-3,1] => [1,-3,4,-2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,-2,4,-3,-1] => [-1,-3,4,-2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[-5,-2,-4,3,1] => [1,3,-4,-2,-5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
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Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Map
reverse
Description
The reversal of a signed permutation.
Map
permutation
Description
The permutation obtained by forgetting the colours.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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