Identifier
-
Mp00090:
Permutations
—cycle-as-one-line notation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
[1] => [1] => [1] => ([],1) => 0
[1,2] => [1,2] => [2,1] => ([],2) => 0
[2,1] => [1,2] => [2,1] => ([],2) => 0
[1,2,3] => [1,2,3] => [3,2,1] => ([],3) => 0
[1,3,2] => [1,2,3] => [3,2,1] => ([],3) => 0
[2,1,3] => [1,2,3] => [3,2,1] => ([],3) => 0
[2,3,1] => [1,2,3] => [3,2,1] => ([],3) => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4) => 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => ([],4) => 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => ([],4) => 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => ([],4) => 0
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4) => 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => ([],4) => 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => ([],4) => 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([],4) => 0
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4) => 0
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4) => 0
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4) => 0
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4) => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 0
[2,4,1,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[2,4,3,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[3,1,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[3,2,4,1,5] => [1,3,4,2,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[3,5,4,1,2] => [1,3,4,2,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
[3,5,4,2,1] => [1,3,4,2,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
[4,1,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 0
[4,1,2,5,3] => [1,4,5,3,2] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 1
[4,1,5,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5) => 0
[4,2,1,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 0
[4,2,1,5,3] => [1,4,5,3,2] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 1
[4,2,5,3,1] => [1,4,3,5,2] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5) => 0
[4,5,1,3,2] => [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 0
[4,5,2,3,1] => [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 0
[5,1,2,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 0
[5,2,1,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,2,5,3,6,4] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[1,2,5,4,6,3] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 0
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[1,3,5,2,6,4] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[1,3,5,4,6,2] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 0
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 0
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6) => 0
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6) => 0
[1,4,6,5,2,3] => [1,2,4,5,3,6] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[1,4,6,5,3,2] => [1,2,4,5,3,6] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6) => 0
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [6,5,2,1,3,4] => ([(2,5),(3,5),(5,4)],6) => 1
[1,5,2,6,4,3] => [1,2,5,4,6,3] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6) => 0
[1,5,3,2,4,6] => [1,2,5,4,3,6] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6) => 0
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [6,5,2,1,3,4] => ([(2,5),(3,5),(5,4)],6) => 1
[1,5,3,6,4,2] => [1,2,5,4,6,3] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6) => 0
[1,5,6,2,4,3] => [1,2,5,4,3,6] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6) => 0
[1,5,6,3,4,2] => [1,2,5,4,3,6] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6) => 0
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6) => 0
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6) => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 0
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Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
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 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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