Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1]=>0 [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,4,1,3]=>000 [(1,3),(2,4)]=>[3,4,1,2]=>[4,3,2,1]=>[4,3,2,1]=>000 [(1,4),(2,3)]=>[4,3,2,1]=>[4,3,2,1]=>[4,3,2,1]=>000 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,4,6,1,3,5]=>00000 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,3,2,1,6,5]=>[4,3,2,6,1,5]=>00000 [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,3,2,1,6,5]=>[4,3,2,6,1,5]=>00000 [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,5,3,4,2,1]=>[3,6,5,4,2,1]=>00000 [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[6,5,3,4,2,1]=>[3,6,5,4,2,1]=>00000 [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>00000 [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>00000 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>00000 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,4,1,2]=>[5,3,1,6,4,2]=>00000 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,5,4,3]=>[6,5,2,4,1,3]=>00000 [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,6,5,4,3]=>[6,5,2,4,1,3]=>00000 [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,5,3,4,2,1]=>[3,6,5,4,2,1]=>00000 [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>00000 [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>00000 [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>00000 [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[2,4,6,8,1,3,5,7]=>0000000 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>[4,3,2,6,8,1,5,7]=>0000000 [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>[4,3,2,6,8,1,5,7]=>0000000 [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[6,5,3,4,2,1,8,7]=>[3,6,5,4,2,8,1,7]=>0000000 [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[6,5,3,4,2,1,8,7]=>[3,6,5,4,2,8,1,7]=>0000000 [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,8,1,7]=>0000000 [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,8,1,7]=>0000000 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,8,1,7]=>0000000 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,6,3,4,1,2,8,7]=>[5,3,1,6,4,8,2,7]=>0000000 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,5,4,3,8,7]=>[6,5,2,4,8,1,3,7]=>0000000 [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,6,5,4,3,8,7]=>[6,5,2,4,8,1,3,7]=>0000000 [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[6,5,3,4,2,1,8,7]=>[3,6,5,4,2,8,1,7]=>0000000 [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,8,1,7]=>0000000 [(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,8,1,7]=>0000000 [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,8,1,7]=>0000000 [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[8,7,6,4,5,3,2,1]=>[4,8,7,6,5,3,2,1]=>0000000 [(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[8,7,6,4,5,3,2,1]=>[4,8,7,6,5,3,2,1]=>0000000 [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,8,7,5,6,4,3]=>[5,8,7,6,2,4,1,3]=>0000000 [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,8,7,5,6,4,3]=>[5,8,7,6,2,4,1,3]=>0000000 [(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[8,7,6,4,5,3,2,1]=>[4,8,7,6,5,3,2,1]=>0000000 [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[8,7,3,6,5,4,2,1]=>[8,7,3,6,5,4,2,1]=>0000000 [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,8,7,6,5,4,3]=>[8,7,6,5,2,4,1,3]=>0000000 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,8,7,6,5,4,3]=>[8,7,6,5,2,4,1,3]=>0000000 [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[7,8,3,6,5,4,1,2]=>[7,8,3,1,6,5,4,2]=>0000000 [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[7,8,6,4,5,3,1,2]=>[7,4,1,8,6,5,3,2]=>0000000 [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[7,8,6,4,5,3,1,2]=>[7,4,1,8,6,5,3,2]=>0000000 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[6,8,3,7,5,1,4,2]=>[3,6,8,1,7,5,4,2]=>0000000 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,7,6,5,4,3]=>[8,7,6,5,2,4,1,3]=>0000000 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,7,8,5,6,3,4]=>[7,5,8,6,2,1,3,4]=>0000000 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,8,1,6,2,4]=>[5,7,3,1,8,2,6,4]=>0000000 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[7,5,8,4,2,6,1,3]=>[5,2,7,4,1,8,6,3]=>0000000 [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[7,5,8,4,2,6,1,3]=>[5,2,7,4,1,8,6,3]=>0000000 [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,7,6,4,5,3,2,1]=>[4,8,7,6,5,3,2,1]=>0000000 [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[8,7,6,4,5,3,2,1]=>[4,8,7,6,5,3,2,1]=>0000000 [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[8,7,3,6,5,4,2,1]=>[8,7,3,6,5,4,2,1]=>0000000 [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[8,7,3,6,5,4,2,1]=>[8,7,3,6,5,4,2,1]=>0000000 [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[7,5,3,8,2,6,1,4]=>[3,5,2,7,1,8,6,4]=>0000000 [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>[4,8,3,7,2,6,1,5]=>0000000 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>[4,8,3,7,2,6,1,5]=>0000000 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,7,6,5]=>[8,7,2,4,6,1,3,5]=>0000000 [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,8,7,6,5]=>[8,7,2,4,6,1,3,5]=>0000000 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,3,2,1,8,7,6,5]=>[4,8,3,7,2,6,1,5]=>0000000 [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[4,3,2,1,8,7,6,5]=>[4,8,3,7,2,6,1,5]=>0000000 [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[8,5,3,7,2,6,4,1]=>[3,5,8,2,7,6,4,1]=>0000000 [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[8,7,3,6,5,4,2,1]=>[8,7,3,6,5,4,2,1]=>0000000 [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[8,7,3,6,5,4,2,1]=>[8,7,3,6,5,4,2,1]=>0000000 [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[8,5,7,4,2,6,3,1]=>[2,8,5,4,7,6,3,1]=>0000000 [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[8,5,7,4,2,6,3,1]=>[2,8,5,4,7,6,3,1]=>0000000 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,7,1,6,4,2]=>[5,8,3,7,1,6,4,2]=>0000000 [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,7,5,6,4,3]=>[5,8,7,6,2,4,1,3]=>0000000 [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,8,7,6,5,4,3]=>[8,7,6,5,2,4,1,3]=>0000000 [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[6,8,3,7,5,1,4,2]=>[3,6,8,1,7,5,4,2]=>0000000 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,7,6,4,5,3,2,1]=>[4,8,7,6,5,3,2,1]=>0000000 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[8,7,6,4,5,3,2,1]=>[4,8,7,6,5,3,2,1]=>0000000 [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[7,8,3,6,5,4,1,2]=>[7,8,3,1,6,5,4,2]=>0000000 [(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,8,7,6,5,4,3]=>[8,7,6,5,2,4,1,3]=>0000000 [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,8,7,6,5,4,3]=>[8,7,6,5,2,4,1,3]=>0000000 [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[8,7,3,6,5,4,2,1]=>[8,7,3,6,5,4,2,1]=>0000000 [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[8,7,6,4,5,3,2,1]=>[4,8,7,6,5,3,2,1]=>0000000 [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000 [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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 $\pi = \pi^{-1}$.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
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.