Identifier
Mp00058: Perfect matchings to permutationPermutations
Mp00065: Permutations permutation posetPosets
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)
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 $y \leq x$.