Identifier
-
Mp00068:
Permutations
—Simion-Schmidt map⟶
Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
[1] => [1] => [1] => ([],1) => 0
[1,2] => [1,2] => [2,1] => ([(0,1)],2) => 0
[2,1] => [2,1] => [1,2] => ([(0,1)],2) => 0
[1,2,3] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[2,1,3] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[3,2,1] => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,5,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,5,2,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,2,3,4,5,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,3,6,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,3,6,5,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,4,3,5,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,4,3,6,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,4,5,3,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,4,5,6,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,4,6,3,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,4,6,5,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,5,3,4,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,5,3,6,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,5,4,3,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,5,4,6,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,5,6,3,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,5,6,4,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,6,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,6,3,5,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,6,4,3,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,6,4,5,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,6,5,3,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,2,6,5,4,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,2,4,5,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,2,4,6,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,2,5,4,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,2,5,6,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,2,6,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,2,6,5,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,4,2,5,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,4,2,6,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,4,5,2,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,4,5,6,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,4,6,2,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,4,6,5,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,5,2,4,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,5,2,6,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,5,4,2,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,5,4,6,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,5,6,2,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,5,6,4,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,6,2,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,6,2,5,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,6,4,2,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,6,4,5,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,6,5,2,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,2,3,5,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,2,3,6,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,2,5,3,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,2,5,6,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,2,6,3,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,2,6,5,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,3,2,5,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,3,2,6,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,3,5,2,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,3,5,6,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,3,6,2,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,3,6,5,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,5,2,3,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,4,5,2,6,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],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 Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
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].
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