Identifier
-
Mp00064:
Permutations
—reverse⟶
Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001711: Integer partitions ⟶ ℤ
Values
[1] => [1] => ([],1) => [1] => 1
[1,2] => [2,1] => ([],2) => [2] => 1
[2,1] => [1,2] => ([(0,1)],2) => [1] => 1
[1,2,3] => [3,2,1] => ([],3) => [3,3] => 1
[1,3,2] => [2,3,1] => ([(1,2)],3) => [3] => 1
[2,1,3] => [3,1,2] => ([(1,2)],3) => [3] => 1
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3) => [2] => 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3) => [2] => 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => [1] => 1
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4) => [8] => 1
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4) => [8] => 1
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4) => [4] => 1
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4) => [4,2] => 1
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4) => [8] => 1
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => [3,3] => 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => [3,2] => 1
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => [3] => 1
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4) => [8] => 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => [3,2] => 1
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4) => [4] => 1
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => [3] => 1
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => 1
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => [2] => 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => [3,3] => 1
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => [3] => 1
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => [3] => 1
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => [2] => 1
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => [2] => 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => [1] => 1
[1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => [5] => 1
[2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => [8] => 1
[2,5,3,4,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => [8] => 1
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => [7] => 1
[2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => [4] => 1
[3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => [4,2] => 1
[3,4,2,5,1] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5) => [8] => 1
[3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => [3,3] => 1
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [5,3] => 1
[3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5) => [7] => 1
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => 1
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [6] => 3
[3,5,4,2,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => [3] => 1
[4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => [7] => 1
[4,2,3,5,1] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => [8] => 1
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [5,3] => 1
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => 1
[4,3,1,5,2] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => [7] => 1
[4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => [5] => 1
[4,3,2,5,1] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => [4] => 1
[4,3,5,1,2] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [6] => 3
[4,3,5,2,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => [3] => 1
[4,5,1,3,2] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [6] => 3
[4,5,2,1,3] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [6] => 3
[4,5,2,3,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [2,2] => 1
[4,5,3,1,2] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => [2,2] => 1
[4,5,3,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => [2] => 1
[5,1,3,4,2] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [8] => 1
[5,1,4,2,3] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => [8] => 1
[5,1,4,3,2] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4] => 1
[5,2,1,4,3] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => [4,2] => 1
[5,2,3,1,4] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [8] => 1
[5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [3,3] => 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => 1
[5,2,4,3,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [3] => 1
[5,3,1,2,4] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => [8] => 1
[5,3,1,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => 1
[5,3,2,1,4] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4] => 1
[5,3,2,4,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [3] => 1
[5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [2,2] => 1
[5,3,4,2,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [2] => 1
[5,4,1,2,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => [3,3] => 1
[5,4,1,3,2] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [3] => 1
[5,4,2,1,3] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [3] => 1
[5,4,2,3,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [2] => 1
[5,4,3,1,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => [2] => 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => [1] => 1
[1,6,5,4,3,2] => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6) => [6] => 3
[2,6,5,4,3,1] => [1,3,4,5,6,2] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [5] => 1
[3,5,6,4,2,1] => [1,2,4,6,5,3] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => [8] => 1
[3,6,4,5,2,1] => [1,2,5,4,6,3] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [8] => 1
[3,6,5,2,4,1] => [1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6) => [7] => 1
[3,6,5,4,1,2] => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => [4,4] => 1
[3,6,5,4,2,1] => [1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [4] => 1
[4,3,6,5,2,1] => [1,2,5,6,3,4] => ([(0,5),(3,2),(4,1),(5,3),(5,4)],6) => [4,2] => 1
[4,5,3,6,2,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => [8] => 1
[4,5,6,3,2,1] => [1,2,3,6,5,4] => ([(0,4),(4,5),(5,1),(5,2),(5,3)],6) => [3,3] => 1
[4,6,2,5,3,1] => [1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6) => [5,3] => 1
[4,6,3,2,5,1] => [1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6) => [7] => 1
[4,6,3,5,2,1] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [3,2] => 1
[4,6,5,2,3,1] => [1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [6] => 3
[4,6,5,3,1,2] => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => [6] => 3
[4,6,5,3,2,1] => [1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [3] => 1
[5,2,6,4,3,1] => [1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6) => [7] => 1
[5,3,4,6,2,1] => [1,2,6,4,3,5] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [8] => 1
[5,3,6,2,4,1] => [1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6) => [5,3] => 1
[5,3,6,4,2,1] => [1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [3,2] => 1
[5,4,2,6,3,1] => [1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6) => [7] => 1
[5,4,3,2,1,6] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6) => [6] => 3
[5,4,3,2,6,1] => [1,6,2,3,4,5] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [5] => 1
[5,4,3,6,1,2] => [2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => [4,4] => 1
[5,4,3,6,2,1] => [1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [4] => 1
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Description
The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation.
Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that
$$ \alpha\pi\alpha^{-1} = \pi^2.$$
Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that
$$ \alpha\pi\alpha^{-1} = \pi^2.$$
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
promotion cycle type
Description
The cycle type of promotion on the linear extensions of a poset.
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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