Identifier
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>1 [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>101 [(1,3),(2,4)]=>[3,4,1,2]=>[4,1,3,2]=>110 [(1,4),(2,3)]=>[3,4,2,1]=>[4,2,3,1]=>110 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>10101 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>11001 [(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[4,2,3,1,6,5]=>11001 [(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[6,2,3,1,5,4]=>11010 [(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[6,4,5,2,3,1]=>11010 [(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[6,2,5,3,4,1]=>11100 [(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[6,2,5,1,4,3]=>11100 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,5,2,4,3]=>11100 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[6,1,3,2,5,4]=>11010 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>10110 [(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[2,1,6,4,5,3]=>10110 [(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[6,4,5,1,3,2]=>11010 [(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[6,1,5,3,4,2]=>11100 [(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[6,3,5,1,4,2]=>11100 [(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[6,3,5,2,4,1]=>11100 [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>1010101 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>1100101 [(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[4,2,3,1,6,5,8,7]=>1100101 [(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[6,2,3,1,5,4,8,7]=>1101001 [(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[6,4,5,2,3,1,8,7]=>1101001 [(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[8,4,5,2,3,1,7,6]=>1101010 [(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[8,6,7,4,5,2,3,1]=>1101010 [(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[8,6,7,2,5,3,4,1]=>1110010 [(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[8,2,5,3,4,1,7,6]=>1110010 [(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[6,2,5,3,4,1,8,7]=>1110001 [(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[6,2,5,1,4,3,8,7]=>1110001 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,5,2,4,3,8,7]=>1110001 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[6,1,3,2,5,4,8,7]=>1101001 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>1011001 [(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[2,1,6,4,5,3,8,7]=>1011001 [(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[6,4,5,1,3,2,8,7]=>1101001 [(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[6,1,5,3,4,2,8,7]=>1110001 [(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[6,3,5,1,4,2,8,7]=>1110001 [(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[6,3,5,2,4,1,8,7]=>1110001 [(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[8,3,5,2,4,1,7,6]=>1110010 [(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[8,6,7,3,5,2,4,1]=>1110010 [(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[8,3,7,5,6,2,4,1]=>1110100 [(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[8,3,7,2,4,1,6,5]=>1110100 [(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[8,3,7,1,4,2,6,5]=>1110100 [(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[8,3,5,1,4,2,7,6]=>1110010 [(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[8,1,5,3,4,2,7,6]=>1110010 [(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[8,4,5,1,3,2,7,6]=>1101010 [(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[2,1,8,4,5,3,7,6]=>1011010 [(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[2,1,8,6,7,4,5,3]=>1011010 [(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[8,6,7,4,5,1,3,2]=>1101010 [(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[8,6,7,1,5,3,4,2]=>1110010 [(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[8,6,7,3,5,1,4,2]=>1110010 [(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[8,3,7,5,6,1,4,2]=>1110100 [(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[8,5,7,3,6,1,4,2]=>1110100 [(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[8,5,7,3,6,2,4,1]=>1110100 [(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[8,3,7,4,6,2,5,1]=>1111000 [(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[8,3,7,4,6,1,5,2]=>1111000 [(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[8,3,7,1,6,4,5,2]=>1111000 [(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[8,1,7,3,6,4,5,2]=>1111000 [(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[8,1,7,5,6,3,4,2]=>1110100 [(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[8,4,7,5,6,1,3,2]=>1101100 [(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[2,1,8,4,7,5,6,3]=>1011100 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[2,1,8,4,7,3,6,5]=>1011100 [(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[8,4,7,1,3,2,6,5]=>1101100 [(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[8,1,7,3,4,2,6,5]=>1110100 [(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[8,1,7,3,6,2,5,4]=>1111000 [(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[8,3,7,1,6,2,5,4]=>1111000 [(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[8,3,7,2,6,1,5,4]=>1111000 [(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[8,3,7,2,6,4,5,1]=>1111000 [(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[8,2,7,3,6,4,5,1]=>1111000 [(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[8,2,7,3,6,1,5,4]=>1111000 [(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[8,2,7,1,6,3,5,4]=>1111000 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,7,2,6,3,5,4]=>1111000 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[8,1,7,2,4,3,6,5]=>1110100 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[8,1,3,2,7,4,6,5]=>1101100 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,7,4,6,5]=>1011100 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,8,3,5,4,7,6]=>1011010 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[8,1,3,2,5,4,7,6]=>1101010 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[8,1,5,2,4,3,7,6]=>1110010 [(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[8,2,5,1,4,3,7,6]=>1110010 [(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[8,2,7,1,4,3,6,5]=>1110100 [(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[8,2,7,3,4,1,6,5]=>1110100 [(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[8,2,7,5,6,3,4,1]=>1110100 [(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[8,4,7,5,6,2,3,1]=>1101100 [(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[8,4,7,2,3,1,6,5]=>1101100 [(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[8,2,3,1,7,4,6,5]=>1101100 [(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[8,2,3,1,5,4,7,6]=>1101010 [(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[4,2,3,1,8,5,7,6]=>1100110 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>1100110 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>1010110 [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[2,1,4,3,8,6,7,5]=>1010110 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[4,1,3,2,8,6,7,5]=>1100110 [(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[4,2,3,1,8,6,7,5]=>1100110 [(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[8,6,7,2,3,1,5,4]=>1101010 [(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[8,2,3,1,7,5,6,4]=>1101100 [(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[8,5,7,2,3,1,6,4]=>1101100 [(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[8,5,7,4,6,2,3,1]=>1101100 [(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[8,5,7,2,6,3,4,1]=>1110100 [(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[8,5,7,2,6,1,4,3]=>1110100 [(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[8,2,7,5,6,1,4,3]=>1110100 [(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[8,6,7,2,5,1,4,3]=>1110010 [(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[8,6,7,1,5,2,4,3]=>1110010 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[8,6,7,1,3,2,5,4]=>1101010 [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[2,1,8,6,7,3,5,4]=>1011010 [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[2,1,8,3,7,5,6,4]=>1011100 [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[8,1,3,2,7,5,6,4]=>1101100 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[8,1,7,5,6,2,4,3]=>1110100 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[8,1,7,2,6,4,5,3]=>1111000 [(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[8,2,7,1,6,4,5,3]=>1111000 [(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[8,2,7,4,6,1,5,3]=>1111000 [(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[8,2,7,4,6,3,5,1]=>1111000 [(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[8,4,7,2,6,3,5,1]=>1111000 [(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[8,4,7,2,6,1,5,3]=>1111000 [(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[8,4,7,1,6,2,5,3]=>1111000 [(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[8,1,7,4,6,2,5,3]=>1111000 [(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[8,5,7,1,6,2,4,3]=>1110100 [(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[8,5,7,1,3,2,6,4]=>1101100 [(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[2,1,8,5,7,3,6,4]=>1011100 [(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[2,1,8,5,7,4,6,3]=>1011100 [(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[8,5,7,4,6,1,3,2]=>1101100 [(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[8,5,7,1,6,3,4,2]=>1110100 [(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[8,1,7,4,6,3,5,2]=>1111000 [(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[8,4,7,1,6,3,5,2]=>1111000 [(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[8,4,7,3,6,1,5,2]=>1111000 [(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[8,4,7,3,6,2,5,1]=>1111000 [(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[2,1,4,3,6,5,8,7,10,9]=>101010101 [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>10101010101
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
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
descent bottoms
Description
The descent bottoms of a permutation as a binary word.