Identifier
-
Mp00253:
Decorated permutations
—permutation⟶
Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤ
Values
[+] => [1] => ([],1) => ([(0,1)],2) => 1
[-] => [1] => ([],1) => ([(0,1)],2) => 1
[+,+] => [1,2] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 1
[-,+] => [1,2] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 1
[+,-] => [1,2] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 1
[-,-] => [1,2] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 1
[2,1] => [2,1] => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[+,+,+] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[-,+,+] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[+,-,+] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[+,+,-] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[-,-,+] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[-,+,-] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[+,-,-] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[-,-,-] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[+,3,2] => [1,3,2] => ([(0,1),(0,2)],3) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 2
[-,3,2] => [1,3,2] => ([(0,1),(0,2)],3) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 2
[2,1,+] => [2,1,3] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[2,1,-] => [2,1,3] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[2,3,1] => [2,3,1] => ([(1,2)],3) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[3,1,2] => [3,1,2] => ([(1,2)],3) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[+,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[-,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[+,-,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[+,+,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[+,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[-,-,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[-,+,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[-,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[+,-,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[+,-,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[+,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[-,-,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[-,-,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[-,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[+,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[-,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[+,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 2
[-,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 2
[+,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 2
[-,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 2
[+,3,2,+] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 2
[-,3,2,+] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 2
[+,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 2
[-,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 2
[+,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 2
[-,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 2
[+,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 2
[-,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 2
[2,1,+,+] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[2,1,-,+] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[2,1,+,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[2,1,-,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 2
[2,3,1,+] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[2,3,1,-] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[3,1,2,+] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[3,1,2,-] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[+,+,+,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,+,+,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,-,+,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,+,-,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,+,+,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,+,+,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,-,+,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,+,-,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,+,+,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,+,+,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,-,-,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,-,+,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,-,+,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,+,-,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,+,-,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,+,+,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,-,-,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,-,+,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,-,+,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,+,-,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,+,-,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,+,+,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,-,-,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,-,-,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,-,+,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,+,-,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,-,-,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,-,-,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,-,+,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,+,-,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,-,-,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[-,-,-,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[+,+,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[-,+,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[+,-,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[+,+,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[-,-,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[-,+,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[+,-,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[-,-,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[+,+,4,3,+] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 2
[-,+,4,3,+] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 2
[+,-,4,3,+] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 2
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Description
The breadth of a lattice.
The breadth of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
The breadth of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
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
permutation
Description
The underlying permutation of the decorated permutation.
Map
order ideals
Description
The lattice of order ideals of a poset.
An order ideal $\mathcal I$ in a poset $P$ is a downward closed set, i.e., $a \in \mathcal I$ and $b \leq a$ implies $b \in \mathcal I$. This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.
An order ideal $\mathcal I$ in a poset $P$ is a downward closed set, i.e., $a \in \mathcal I$ and $b \leq a$ implies $b \in \mathcal I$. This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.
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