Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
Images
=>
Cc0012;cc-rep-0Cc0014;cc-rep-2Cc0014;cc-rep-3
[(1,2)]=>[2,1]=>([],2)=>([],2) [(1,2),(3,4)]=>[2,1,4,3]=>([(0,2),(0,3),(1,2),(1,3)],4)=>([(0,2),(0,3),(1,2),(1,3)],4) [(1,3),(2,4)]=>[3,4,1,2]=>([(0,3),(1,2)],4)=>([(0,3),(1,2)],4) [(1,4),(2,3)]=>[4,3,2,1]=>([],4)=>([],4) [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)=>([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)=>([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6) [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6) [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>([(2,4),(2,5),(3,4),(3,5)],6)=>([(2,4),(2,5),(3,4),(3,5)],6) [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>([(2,5),(3,4)],6)=>([(2,5),(3,4)],6) [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>([(0,5),(1,5),(2,4),(3,4)],6)=>([(0,4),(0,5),(1,2),(1,3)],6) [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>([(0,5),(1,4),(4,2),(5,3)],6)=>([(0,5),(1,4),(4,2),(5,3)],6) [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)=>([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6) [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)=>([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6) [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)=>([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>([(0,4),(0,5),(1,2),(1,3)],6)=>([(0,5),(1,5),(2,4),(3,4)],6) [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>([(2,5),(3,4)],6)=>([(2,5),(3,4)],6) [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>([],6)=>([],6) [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)=>([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8) [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)=>([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8) [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)=>([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8) [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8) [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)=>([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8) [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,3),(7,2)],8) [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)=>([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8) [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)=>([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8) [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)=>([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8) [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) [(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,3),(7,2)],8) [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8) [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)=>([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8) [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>([(4,6),(4,7),(5,6),(5,7)],8)=>([(4,6),(4,7),(5,6),(5,7)],8) [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>([(4,7),(5,6)],8)=>([(4,7),(5,6)],8) [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,3),(7,2)],8)=>([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8) [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)=>([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8) [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)=>([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8) [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)=>([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8) [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)=>([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8) [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)=>([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8) [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)=>([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8) [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)=>([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8) [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(6,5),(7,4)],8)=>([(0,3),(0,7),(1,2),(1,6),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6)],8) [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)=>([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8) [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)=>([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8) [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)=>([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8) [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)=>([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>([(4,7),(5,6)],8)=>([(4,7),(5,6)],8) [(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,3),(7,2)],8)=>([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8) [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)=>([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7)],8)=>([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7)],8)=>([(0,7),(1,7),(2,6),(3,6),(4,6),(4,7),(5,6),(5,7)],8) [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4)],8)=>([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8) [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>([(4,7),(5,6)],8)=>([(4,7),(5,6)],8) [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>([],8)=>([],8)
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
dual poset
Description
The dual of a poset.
The dual (or opposite) of a poset $(\mathcal P,\leq)$ is the poset $(\mathcal P^d,\leq_d)$ with $x \leq_d y$ if $y \leq x$.