Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>1=>1 [(1,2),(3,4)]=>[2,1,4,3]=>101=>101 [(1,3),(2,4)]=>[3,4,1,2]=>010=>010 [(1,4),(2,3)]=>[4,3,2,1]=>111=>111 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>10101=>10101 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>01001=>10010 [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>11101=>10111 [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>11010=>01011 [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>11011=>11011 [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>10101=>10101 [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>10110=>01101 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>00100=>00100 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>01010=>01010 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>10010=>01001 [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>10111=>11101 [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>01011=>11010 [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>01101=>10110 [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>01110=>01110 [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>11111=>11111 [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>1010101=>1010101 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>0100101=>1010010 [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>1110101=>1010111 [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>1101001=>1001011 [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>1101101=>1011011 [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>1101010=>0101011 [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>1101011=>1101011 [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>1010011=>1100101 [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>1010010=>0100101 [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>1010101=>1010101 [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>1011001=>1001101 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>0010001=>1000100 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>0101001=>1001010 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>1001001=>1001001 [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>1011101=>1011101 [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>0101101=>1011010 [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>0110101=>1010110 [(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>0111001=>1001110 [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>1111101=>1011111 [(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>1111010=>0101111 [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>1111011=>1101111 [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>1110101=>1010111 [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>1110110=>0110111 [(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>0110100=>0010110 [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>0111010=>0101110 [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>0110010=>0100110 [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>0101010=>0101010 [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>1011010=>0101101 [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>1011011=>1101101 [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>0101011=>1101010 [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>0110011=>1100110 [(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>0111011=>1101110 [(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>0110101=>1010110 [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>0110110=>0110110 [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>1110111=>1110111 [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>1101011=>1101011 [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>0101010=>0101010 [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>0101101=>1011010 [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>0101001=>1001010 [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>0110101=>1010110 [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>0100101=>1010010 [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>1010101=>1010101 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>1010110=>0110101 [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>0100110=>0110010 [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>0110110=>0110110 [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>0101010=>0101010 [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>0101100=>0011010 [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>1101110=>0111011 [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>1101101=>1011011 [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>1001001=>1001001 [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>1001010=>0101001 [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>1001100=>0011001 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>0001000=>0001000 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>0010100=>0010100 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>0100100=>0010010 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>1000100=>0010001 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>1001010=>0101001 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>0101010=>0101010 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>0010010=>0100100 [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>1011010=>0101101 [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>1010100=>0010101 [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>1010110=>0110101 [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>1010101=>1010101 [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>1100101=>1010011 [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>1100110=>0110011 [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>1100100=>0010011 [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>1101010=>0101011 [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>1110010=>0100111 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>0100010=>0100010 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>1010010=>0100101 [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>1010111=>1110101 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>0100111=>1110010 [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>1110111=>1110111 [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>1101011=>1101011 [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>1101101=>1011011 [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>1101110=>0111011 [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>1101111=>1111011 [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>1010111=>1110101 [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>1010110=>0110101 [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>1010101=>1010101 [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>1011011=>1101101 [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>0010011=>1100100 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>0101011=>1101010 [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>1001011=>1101001 [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>1001101=>1011001 [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>0101101=>1011010 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>0010101=>1010100 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>0011001=>1001100 [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>1011101=>1011101 [(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>1011010=>0101101 [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>1011011=>1101101 [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>1011101=>1011101 [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>1011110=>0111101 [(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>0011100=>0011100 [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>0011010=>0101100 [(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>0010110=>0110100 [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>0101110=>0111010 [(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>1001110=>0111001 [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>1011111=>1111101 [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>0101111=>1111010 [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>0110111=>1110110 [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>0111011=>1101110 [(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>0111101=>1011110 [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>0111110=>0111110 [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>1111111=>1111111 [(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>101010101=>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]=>10101010101=>10101010101
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.
Map
reverse
Description
Return the reversal of a binary word.