Identifier
-
Mp00159:
Permutations
—Demazure product with inverse⟶
Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤ
Values
[1] => [1] => [1] => ([],1) => 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2) => 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2) => 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,3,1,4] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,1,2,4] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,2,1,4] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,4,6,5,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,5,6,3,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,5,6,4,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,6,4,3,5,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,6,4,5,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,6,5,3,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,6,5,4,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,4,6,5,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,4,6,5,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,5,6,1,4,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,5,6,2,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,5,6,4,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,5,6,4,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,6,4,1,5,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,6,4,2,5,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,6,4,5,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,6,4,5,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,6,5,1,4,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,6,5,2,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,6,5,4,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,6,5,4,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,2,6,5,1,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,2,6,5,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,3,6,5,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,3,6,5,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,5,6,1,3,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,5,6,2,1,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,5,6,2,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,5,6,3,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,5,6,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,6,2,1,5,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,6,2,3,5,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
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Description
The length of the shortest maximal antichain in a poset.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
See Mp00067Foata bijection.
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.
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