Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>0 [(1,2),(3,4)]=>[2,1,4,3]=>[3,2,4,1]=>000 [(1,3),(2,4)]=>[3,4,1,2]=>[1,4,2,3]=>100 [(1,4),(2,3)]=>[4,3,2,1]=>[4,3,2,1]=>000 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[4,3,5,2,6,1]=>00000 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[2,5,3,4,6,1]=>00000 [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[5,4,3,2,6,1]=>00000 [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[1,5,4,6,2,3]=>10000 [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[6,4,2,5,3,1]=>00000 [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[6,1,5,2,4,3]=>00000 [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[5,1,6,3,2,4]=>00000 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[1,2,6,3,4,5]=>11000 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[4,1,3,6,2,5]=>00000 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[3,2,1,6,4,5]=>00100 [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[5,4,6,3,2,1]=>00000 [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,6,4,5,2,1]=>00000 [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[2,6,5,3,4,1]=>00000 [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[1,6,5,4,2,3]=>10000 [(1,6),(2,5),(3,4)]=>[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]=>[5,4,6,3,7,2,8,1]=>0000000 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,6,4,5,7,2,8,1]=>0000000 [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[6,5,4,3,7,2,8,1]=>0000000 [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[2,6,5,7,3,4,8,1]=>0000000 [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[1,6,4,7,5,8,2,3]=>1000000 [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[1,3,7,4,6,8,2,5]=>1000000 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[2,3,7,4,5,6,8,1]=>0000000 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,2,4,7,3,6,8,1]=>0000000 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[4,3,2,7,5,6,8,1]=>0000000 [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[8,1,6,5,7,2,4,3]=>0000000 [(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[1,2,7,6,8,3,4,5]=>1100000 [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,1,6,7,5,8,2,4]=>0000000 [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[1,8,6,3,7,5,2,4]=>1000000 [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[8,7,5,2,6,4,3,1]=>0000000 [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[8,7,1,6,2,5,4,3]=>0000000 [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[1,8,2,7,3,6,4,5]=>1000000 [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[7,8,1,6,2,4,5,3]=>0000000 [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[2,8,1,7,4,3,6,5]=>0000000 [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,7,4,1,8,5,2,6]=>0000000 [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[6,7,1,4,8,3,2,5]=>0000000 [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[2,7,1,8,4,5,3,6]=>0000000 [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[1,7,2,8,5,3,4,6]=>1000000 [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[7,6,1,8,4,3,2,5]=>0000000 [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[7,1,2,8,3,5,4,6]=>0000000 [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[6,1,2,8,4,3,5,7]=>0000000 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[1,2,3,8,4,5,6,7]=>1110000 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[5,1,2,4,8,3,6,7]=>0000000 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[4,1,3,2,8,5,6,7]=>0001000 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[3,2,1,4,8,5,6,7]=>0011000 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[4,3,6,1,5,8,2,7]=>0000000 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,6,2,1,4,8,3,7]=>0000000 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[2,6,1,4,5,8,3,7]=>0000000 [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[1,6,2,5,8,3,4,7]=>1000000 [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[8,4,2,1,7,3,6,5]=>0000000 [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[7,4,2,1,8,5,3,6]=>0000000 [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[1,5,4,2,8,3,6,7]=>1000000 [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[6,4,3,1,5,8,2,7]=>0000000 [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[5,4,3,2,1,8,6,7]=>0000100 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[2,5,3,4,1,8,6,7]=>0000100 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[4,3,5,2,1,8,6,7]=>0000100 [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[6,5,7,4,8,3,2,1]=>0000000 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,7,5,6,8,3,2,1]=>0000000 [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[7,6,5,4,8,3,2,1]=>0000000 [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,7,6,8,4,5,2,1]=>0000000 [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[1,7,6,8,5,4,2,3]=>1000000 [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[2,4,8,6,7,3,5,1]=>0000000 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[6,3,5,8,4,7,2,1]=>0000000 [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[5,4,3,8,6,7,2,1]=>0000000 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[3,2,8,5,7,4,6,1]=>0000000 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[2,3,8,7,4,5,6,1]=>0000000 [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[7,2,8,6,4,3,5,1]=>0000000 [(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[3,1,8,5,7,6,2,4]=>0000000 [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[5,8,6,7,4,3,2,1]=>0000000 [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[4,8,7,5,6,3,2,1]=>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]=>0000000
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
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.