Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[1,2]=>[2,1]=>0 [(1,2),(3,4)]=>[2,1,4,3]=>[1,2,3,4]=>[2,3,4,1]=>000 [(1,3),(2,4)]=>[3,4,1,2]=>[1,3,2,4]=>[2,4,3,1]=>000 [(1,4),(2,3)]=>[4,3,2,1]=>[1,4,2,3]=>[2,4,1,3]=>000 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[1,2,3,4,5,6]=>[2,3,4,5,6,1]=>00000 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[1,3,2,4,5,6]=>[2,4,3,5,6,1]=>00000 [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[1,4,2,3,5,6]=>[2,4,5,3,6,1]=>00000 [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[1,5,2,3,4,6]=>[2,4,5,6,3,1]=>00000 [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[1,6,2,3,4,5]=>[2,4,5,6,1,3]=>00000 [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[1,6,2,4,3,5]=>[2,4,6,5,1,3]=>00000 [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[1,5,2,4,3,6]=>[2,4,6,5,3,1]=>00000 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[1,4,2,5,3,6]=>[2,4,6,3,5,1]=>00000 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[1,3,2,5,4,6]=>[2,4,3,6,5,1]=>00000 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[1,2,3,5,4,6]=>[2,3,4,6,5,1]=>00000 [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[1,2,3,6,4,5]=>[2,3,4,6,1,5]=>00000 [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[1,3,2,6,4,5]=>[2,4,3,6,1,5]=>00000 [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[1,4,2,6,3,5]=>[2,4,6,3,1,5]=>00000 [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[1,5,2,6,3,4]=>[2,4,6,1,3,5]=>00000 [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[1,6,2,5,3,4]=>[2,4,6,1,5,3]=>00000 [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[1,2,3,4,5,6,7,8]=>[2,3,4,5,6,7,8,1]=>0000000 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[1,3,2,4,5,6,7,8]=>[2,4,3,5,6,7,8,1]=>0000000 [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[1,6,2,4,3,5,7,8]=>[2,4,6,5,7,3,8,1]=>0000000 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[1,4,2,5,3,6,7,8]=>[2,4,6,3,5,7,8,1]=>0000000 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[1,3,2,5,4,6,7,8]=>[2,4,3,6,5,7,8,1]=>0000000 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[1,2,3,5,4,6,7,8]=>[2,3,4,6,5,7,8,1]=>0000000 [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[1,4,2,6,3,5,7,8]=>[2,4,6,3,7,5,8,1]=>0000000 [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[1,2,3,8,4,5,6,7]=>[2,3,4,6,7,8,1,5]=>0000000 [(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[1,5,2,8,3,4,6,7]=>[2,4,6,7,3,8,1,5]=>0000000 [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[1,7,2,8,3,4,5,6]=>[2,4,6,7,8,1,3,5]=>0000000 [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[1,8,2,7,3,4,5,6]=>[2,4,6,7,8,1,5,3]=>0000000 [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[1,6,2,8,3,5,4,7]=>[2,4,6,8,7,3,1,5]=>0000000 [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[1,3,2,8,4,6,5,7]=>[2,4,3,6,8,7,1,5]=>0000000 [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[1,2,3,8,4,6,5,7]=>[2,3,4,6,8,7,1,5]=>0000000 [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[1,7,2,6,3,5,4,8]=>[2,4,6,8,7,5,3,1]=>0000000 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[1,5,2,6,3,7,4,8]=>[2,4,6,8,3,5,7,1]=>0000000 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[1,4,2,6,3,7,5,8]=>[2,4,6,3,8,5,7,1]=>0000000 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[1,3,2,6,4,7,5,8]=>[2,4,3,6,8,5,7,1]=>0000000 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[1,2,3,6,4,7,5,8]=>[2,3,4,6,8,5,7,1]=>0000000 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[1,2,3,5,4,7,6,8]=>[2,3,4,6,5,8,7,1]=>0000000 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[1,3,2,5,4,7,6,8]=>[2,4,3,6,5,8,7,1]=>0000000 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[1,4,2,5,3,7,6,8]=>[2,4,6,3,5,8,7,1]=>0000000 [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[1,5,2,4,3,7,6,8]=>[2,4,6,5,3,8,7,1]=>0000000 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[1,3,2,4,5,7,6,8]=>[2,4,3,5,6,8,7,1]=>0000000 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[1,2,3,4,5,7,6,8]=>[2,3,4,5,6,8,7,1]=>0000000 [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[1,8,2,3,4,7,5,6]=>[2,4,5,6,8,1,7,3]=>0000000 [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[1,7,2,4,3,8,5,6]=>[2,4,6,5,8,1,3,7]=>0000000 [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[1,4,2,5,3,8,6,7]=>[2,4,6,3,5,8,1,7]=>0000000 [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[1,3,2,6,4,8,5,7]=>[2,4,3,6,8,5,1,7]=>0000000 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[1,4,2,6,3,8,5,7]=>[2,4,6,3,8,5,1,7]=>0000000 [(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[1,7,2,5,3,8,4,6]=>[2,4,6,8,5,1,3,7]=>0000000 [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[1,8,2,6,3,7,4,5]=>[2,4,6,8,1,5,7,3]=>0000000 [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[1,7,2,6,3,8,4,5]=>[2,4,6,8,1,5,3,7]=>0000000 [(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[1,6,2,7,3,8,4,5]=>[2,4,6,8,1,3,5,7]=>0000000 [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[1,3,2,7,4,8,5,6]=>[2,4,3,6,8,1,5,7]=>0000000
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
Map
connectivity set
Description
The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.