Identifier
-
Mp00253:
Decorated permutations
—permutation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤ
Values
[3,+,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[3,-,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[4,+,+,1] => [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,-,+,1] => [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,+,-,1] => [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,-,-,1] => [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[5,+,+,+,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[5,-,+,+,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[5,+,-,+,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[5,+,+,-,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[5,-,-,+,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[5,-,+,-,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[5,+,-,-,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[5,-,-,-,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[5,+,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 6
[5,-,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 6
[5,3,2,+,1] => [5,3,2,4,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 6
[5,3,2,-,1] => [5,3,2,4,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 6
[5,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[5,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[5,4,+,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[5,4,-,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[6,+,+,+,+,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,-,+,+,+,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,+,-,+,+,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,+,+,-,+,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,+,+,+,-,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,-,-,+,+,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,-,+,-,+,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,-,+,+,-,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,+,-,-,+,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,+,-,+,-,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,+,+,-,-,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,-,-,-,+,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,-,-,+,-,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,-,+,-,-,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,+,-,-,-,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,-,-,-,-,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[6,+,+,5,4,1] => [6,2,3,5,4,1] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,-,+,5,4,1] => [6,2,3,5,4,1] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,+,-,5,4,1] => [6,2,3,5,4,1] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,-,-,5,4,1] => [6,2,3,5,4,1] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,+,4,3,+,1] => [6,2,4,3,5,1] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,-,4,3,+,1] => [6,2,4,3,5,1] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,+,4,3,-,1] => [6,2,4,3,5,1] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,-,4,3,-,1] => [6,2,4,3,5,1] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,+,4,5,3,1] => [6,2,4,5,3,1] => [1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 10
[6,-,4,5,3,1] => [6,2,4,5,3,1] => [1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 10
[6,+,5,3,4,1] => [6,2,5,3,4,1] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 10
[6,-,5,3,4,1] => [6,2,5,3,4,1] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 10
[6,+,5,+,3,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[6,-,5,+,3,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[6,+,5,-,3,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[6,-,5,-,3,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[6,3,2,+,+,1] => [6,3,2,4,5,1] => [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,3,2,-,+,1] => [6,3,2,4,5,1] => [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,3,2,+,-,1] => [6,3,2,4,5,1] => [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,3,2,-,-,1] => [6,3,2,4,5,1] => [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[6,3,2,5,4,1] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 8
[6,3,4,2,+,1] => [6,3,4,2,5,1] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 10
[6,3,4,2,-,1] => [6,3,4,2,5,1] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 10
[6,3,4,5,2,1] => [6,3,4,5,2,1] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 10
[6,3,5,2,4,1] => [6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 7
[6,3,5,+,2,1] => [6,3,5,4,2,1] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 7
[6,3,5,-,2,1] => [6,3,5,4,2,1] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 7
[6,4,2,3,+,1] => [6,4,2,3,5,1] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 10
[6,4,2,3,-,1] => [6,4,2,3,5,1] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 10
[6,4,2,5,3,1] => [6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 7
[6,4,+,2,+,1] => [6,4,3,2,5,1] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[6,4,-,2,+,1] => [6,4,3,2,5,1] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[6,4,+,2,-,1] => [6,4,3,2,5,1] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[6,4,-,2,-,1] => [6,4,3,2,5,1] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[6,4,+,5,2,1] => [6,4,3,5,2,1] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 7
[6,4,-,5,2,1] => [6,4,3,5,2,1] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 7
[6,4,5,3,2,1] => [6,4,5,3,2,1] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6
[6,5,2,3,4,1] => [6,5,2,3,4,1] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 10
[6,5,2,+,3,1] => [6,5,2,4,3,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 7
[6,5,2,-,3,1] => [6,5,2,4,3,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 7
[6,5,+,2,4,1] => [6,5,3,2,4,1] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 7
[6,5,-,2,4,1] => [6,5,3,2,4,1] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 7
[6,5,+,+,2,1] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[6,5,-,+,2,1] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[6,5,+,-,2,1] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[6,5,-,-,2,1] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[6,5,4,2,3,1] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
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Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
permutation
Description
The underlying permutation of the decorated permutation.
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