Identifier
Mp00283: Perfect matchings —non-nesting-exceedence 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)]=>[3,4,2,1]=>[4,3,1,2]=>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)]=>[3,4,2,1,6,5]=>[5,4,2,3,6,1]=>00000 [(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[1,5,3,6,2,4]=>10000 [(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[6,4,2,5,1,3]=>00000 [(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[6,1,5,2,3,4]=>00000 [(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[5,1,6,2,3,4]=>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)]=>[2,1,5,6,4,3]=>[4,3,6,5,1,2]=>00000 [(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[5,6,2,4,1,3]=>00000 [(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[2,6,5,1,3,4]=>00000 [(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[1,6,5,2,3,4]=>10000 [(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[6,5,4,1,2,3]=>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)]=>[3,4,2,1,6,5,8,7]=>[6,5,3,4,7,2,8,1]=>0000000 [(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[2,6,4,7,3,5,8,1]=>0000000 [(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[7,5,3,6,2,4,8,1]=>0000000 [(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[1,6,4,7,3,8,2,5]=>1000000 [(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[8,5,3,6,2,7,1,4]=>0000000 [(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[8,2,6,3,4,7,1,5]=>0000000 [(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[1,3,7,4,5,8,2,6]=>1000000 [(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[7,2,6,3,4,5,8,1]=>0000000 [(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[6,2,7,3,4,5,8,1]=>0000000 [(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 [(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[5,4,7,6,2,3,8,1]=>0000000 [(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[6,7,3,5,2,4,8,1]=>0000000 [(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[3,7,6,2,4,5,8,1]=>0000000 [(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[2,7,6,3,4,5,8,1]=>0000000 [(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[7,6,5,2,3,4,8,1]=>0000000 [(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[1,7,6,3,4,8,2,5]=>1000000 [(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[8,6,5,2,3,7,1,4]=>0000000 [(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[8,1,6,3,7,2,4,5]=>0000000 [(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[7,1,6,3,8,2,4,5]=>0000000 [(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[1,2,7,4,8,3,5,6]=>1100000 [(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[6,1,7,3,4,8,2,5]=>0000000 [(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[2,1,7,4,5,8,3,6]=>0100000 [(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[5,1,4,7,3,8,2,6]=>0000000 [(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[4,3,1,7,5,8,2,6]=>0000000 [(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[5,4,8,6,2,7,1,3]=>0000000 [(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[6,8,3,5,2,7,1,4]=>0000000 [(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[3,8,6,2,4,7,1,5]=>0000000 [(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[7,8,5,2,3,6,1,4]=>0000000 [(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[2,8,6,3,7,1,4,5]=>0000000 [(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[1,8,6,3,7,2,4,5]=>1000000 [(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[8,7,5,2,6,1,3,4]=>0000000 [(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[8,7,1,6,2,3,4,5]=>0000000 [(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[1,8,2,7,3,4,5,6]=>1000000 [(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[7,8,1,6,2,3,4,5]=>0000000 [(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[2,8,1,7,3,4,5,6]=>0000000 [(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[6,8,1,3,7,2,4,5]=>0000000 [(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[5,8,2,1,7,3,4,6]=>0000000 [(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[4,3,8,1,7,2,5,6]=>0000000 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[4,3,7,1,8,2,5,6]=>0000000 [(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[5,7,2,1,8,3,4,6]=>0000000 [(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[6,7,1,3,8,2,4,5]=>0000000 [(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[2,7,1,8,3,4,5,6]=>0000000 [(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[1,7,2,8,3,4,5,6]=>1000000 [(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[7,6,1,8,2,3,4,5]=>0000000 [(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[8,6,1,7,2,3,4,5]=>0000000 [(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[8,1,2,7,3,4,5,6]=>0000000 [(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[7,1,2,8,3,4,5,6]=>0000000 [(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[6,1,2,8,3,4,5,7]=>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 [(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[6,5,1,3,4,8,2,7]=>0000000 [(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[1,6,2,4,8,3,5,7]=>1000000 [(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[7,5,1,3,8,2,4,6]=>0000000 [(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[8,5,1,3,7,2,4,6]=>0000000 [(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[8,4,2,1,7,3,5,6]=>0000000 [(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[7,4,2,1,8,3,5,6]=>0000000 [(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[1,5,3,2,8,4,6,7]=>1000000 [(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[6,4,2,1,5,8,3,7]=>0000000 [(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[5,4,2,3,1,8,6,7]=>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)]=>[2,1,4,3,7,8,6,5]=>[5,4,6,3,8,7,1,2]=>0000000 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[3,6,4,5,8,7,1,2]=>0000000 [(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[6,5,3,4,8,7,1,2]=>0000000 [(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[7,5,3,8,2,6,1,4]=>0000000 [(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[2,6,4,8,7,1,3,5]=>0000000 [(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[1,6,4,8,7,2,3,5]=>1000000 [(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[8,5,3,7,6,1,2,4]=>0000000 [(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[8,6,7,2,5,1,3,4]=>0000000 [(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[1,7,8,3,6,2,4,5]=>1000000 [(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[2,7,8,3,6,1,4,5]=>0000000 [(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[7,6,8,2,3,5,1,4]=>0000000 [(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[3,7,8,2,4,6,1,5]=>0000000 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[6,7,3,8,2,5,1,4]=>0000000 [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[5,4,7,8,2,6,1,3]=>0000000 [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[4,3,2,8,7,1,5,6]=>0000000 [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[5,2,4,8,7,1,3,6]=>0000000 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[6,2,8,3,7,1,4,5]=>0000000 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[2,3,8,7,1,4,5,6]=>0000000 [(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[7,2,8,6,1,3,4,5]=>0000000 [(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[1,3,8,7,2,4,5,6]=>1000000 [(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[8,2,7,6,1,3,4,5]=>0000000 [(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[8,1,7,6,2,3,4,5]=>0000000 [(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[7,1,8,6,2,3,4,5]=>0000000 [(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[1,2,8,7,3,4,5,6]=>1100000 [(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[2,1,8,7,3,4,5,6]=>0100000 [(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[6,1,8,3,7,2,4,5]=>0000000 [(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[5,1,4,8,7,2,3,6]=>0000000 [(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[4,3,1,8,7,2,5,6]=>0000000 [(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[5,4,8,7,6,1,2,3]=>0000000 [(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[6,8,3,7,5,1,2,4]=>0000000 [(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[7,8,6,2,5,1,3,4]=>0000000 [(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[3,8,7,6,1,2,4,5]=>0000000 [(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[2,8,7,6,1,3,4,5]=>0000000 [(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[1,8,7,6,2,3,4,5]=>1000000 [(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[8,7,6,5,1,2,3,4]=>0000000
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given 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.