Identifier
-
Mp00068:
Permutations
—Simion-Schmidt map⟶
Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
[1] => [1] => [1] => ([],1) => 0
[1,2] => [1,2] => [1,2] => ([(0,1)],2) => 0
[2,1] => [2,1] => [2,1] => ([],2) => 0
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => 0
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3) => 0
[3,2,1] => [3,2,1] => [3,2,1] => ([],3) => 0
[2,1,3,4] => [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 0
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 0
[2,3,1,4] => [2,4,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4) => 0
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4) => 0
[3,1,2,4] => [3,1,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 0
[3,1,4,2] => [3,1,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4) => 0
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 0
[4,2,1,3] => [4,2,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4) => 0
[2,3,1,4,5] => [2,5,1,4,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[2,3,1,5,4] => [2,5,1,4,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[2,3,4,1,5] => [2,5,4,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[2,3,5,1,4] => [2,5,4,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[2,4,1,3,5] => [2,5,1,4,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[2,4,1,5,3] => [2,5,1,4,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[2,4,3,1,5] => [2,5,4,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[2,4,5,1,3] => [2,5,4,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[2,5,1,3,4] => [2,5,1,4,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[2,5,1,4,3] => [2,5,1,4,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[2,5,3,1,4] => [2,5,4,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[2,5,4,1,3] => [2,5,4,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[3,2,1,4,5] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[3,2,4,1,5] => [3,2,5,1,4] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 1
[3,2,5,1,4] => [3,2,5,1,4] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 1
[3,4,1,2,5] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[3,4,1,5,2] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[3,4,2,5,1] => [3,5,2,4,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
[3,5,1,2,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[3,5,2,4,1] => [3,5,2,4,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
[4,2,1,3,5] => [4,2,1,5,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[4,2,1,5,3] => [4,2,1,5,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[4,2,3,1,5] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[4,2,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[4,3,1,2,5] => [4,3,1,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[4,3,1,5,2] => [4,3,1,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,3,5,1,2] => [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 1
[4,3,5,2,1] => [4,3,5,2,1] => [4,3,5,2,1] => ([(2,4),(3,4)],5) => 0
[4,5,1,2,3] => [4,5,1,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[4,5,1,3,2] => [4,5,1,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[4,5,2,1,3] => [4,5,2,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[4,5,3,1,2] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[5,2,1,3,4] => [5,2,1,4,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
[5,2,1,4,3] => [5,2,1,4,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
[5,3,1,2,4] => [5,3,1,4,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[5,3,1,4,2] => [5,3,1,4,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[5,3,2,1,4] => [5,3,2,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[5,3,4,1,2] => [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
[5,4,2,1,3] => [5,4,2,1,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5) => 0
[2,3,4,5,1,6] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,3,5,4,1,6] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,3,5,6,1,4] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,3,6,4,1,5] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,3,6,5,1,4] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,4,3,5,1,6] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,4,3,6,1,5] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,4,5,3,1,6] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,4,5,6,1,3] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,4,6,3,1,5] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,4,6,5,1,3] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,5,3,4,1,6] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,5,3,6,1,4] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,5,4,3,1,6] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,5,4,6,1,3] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,5,6,3,1,4] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,5,6,4,1,3] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,6,3,4,1,5] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,6,3,5,1,4] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,6,4,3,1,5] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,6,4,5,1,3] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,6,5,3,1,4] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[2,6,5,4,1,3] => [2,6,5,4,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[3,4,5,2,6,1] => [3,6,5,2,4,1] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[3,4,6,2,5,1] => [3,6,5,2,4,1] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[3,5,4,2,6,1] => [3,6,5,2,4,1] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[3,5,6,2,4,1] => [3,6,5,2,4,1] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[3,6,4,2,5,1] => [3,6,5,2,4,1] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[3,6,5,2,4,1] => [3,6,5,2,4,1] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[4,5,3,6,2,1] => [4,6,3,5,2,1] => [6,4,3,5,2,1] => ([(3,5),(4,5)],6) => 0
[4,6,3,5,2,1] => [4,6,3,5,2,1] => [6,4,3,5,2,1] => ([(3,5),(4,5)],6) => 0
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[5,4,6,3,2,1] => [5,4,6,3,2,1] => [5,4,6,3,2,1] => ([(3,5),(4,5)],6) => 0
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [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
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
See Mp00067Foata bijection.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
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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