Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[1,2]=>0
[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[3,4,1,2]=>010
[(1,3),(2,4)]=>[3,4,1,2]=>[4,1,3,2]=>[1,4,2,3]=>010
[(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1]=>[2,3,1,4]=>010
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[5,6,3,4,1,2]=>01010
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>[3,6,4,5,1,2]=>01010
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[4,5,3,6,1,2]=>01010
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,6,1,5,4]=>[4,5,1,6,2,3]=>01010
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[4,5,2,3,1,6]=>01010
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[5,2,4,3,6,1]=>[2,5,3,4,1,6]=>01010
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[6,2,4,1,5,3]=>[1,5,3,6,2,4]=>01010
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,5,2,4,3]=>[1,6,2,5,3,4]=>01010
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[6,1,3,2,5,4]=>[1,6,4,5,2,3]=>01010
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>[5,6,1,4,2,3]=>01010
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[5,6,2,3,1,4]=>01010
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[5,4,6,1,3,2]=>[2,3,1,6,4,5]=>01010
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,1,6,3,4,2]=>[2,6,1,4,3,5]=>01010
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,3,6,1,5,2]=>[3,4,1,6,2,5]=>01010
[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[3,4,2,5,1,6]=>01010
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[7,8,5,6,3,4,1,2]=>0101010
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>[5,8,6,7,3,4,1,2]=>0101010
[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7]=>[6,7,5,8,3,4,1,2]=>0101010
[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[3,2,6,1,5,4,8,7]=>[6,7,3,8,4,5,1,2]=>0101010
[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,2,5,4,6,1,8,7]=>[6,7,4,5,3,8,1,2]=>0101010
[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[3,2,5,4,8,1,7,6]=>[6,7,4,5,1,8,2,3]=>0101010
[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,2,5,4,7,6,8,1]=>[6,7,4,5,2,3,1,8]=>0101010
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,5,2,4,3,8,7]=>[3,8,4,7,5,6,1,2]=>0101010
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[6,1,3,2,5,4,8,7]=>[3,8,6,7,4,5,1,2]=>0101010
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>[7,8,3,6,4,5,1,2]=>0101010
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,5,4,6,3,8,7]=>[7,8,4,5,3,6,1,2]=>0101010
[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[5,4,6,1,3,2,8,7]=>[4,5,3,8,6,7,1,2]=>0101010
[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[4,3,6,1,5,2,8,7]=>[5,6,3,8,4,7,1,2]=>0101010
[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7]=>[5,6,4,7,3,8,1,2]=>0101010
[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[4,3,5,2,8,1,7,6]=>[5,6,4,7,1,8,2,3]=>0101010
[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1]=>[5,6,4,7,2,3,1,8]=>0101010
[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[4,3,7,2,6,5,8,1]=>[5,6,2,7,3,4,1,8]=>0101010
[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[5,4,8,1,3,2,7,6]=>[4,5,1,8,6,7,2,3]=>0101010
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,5,4,8,3,7,6]=>[7,8,4,5,1,6,2,3]=>0101010
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,5,4,7,6,8,3]=>[7,8,4,5,2,3,1,6]=>0101010
[(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[5,4,7,6,8,1,3,2]=>[4,5,2,3,1,8,6,7]=>0101010
[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[4,3,7,6,8,1,5,2]=>[5,6,2,3,1,8,4,7]=>0101010
[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[4,3,6,5,8,1,7,2]=>[5,6,3,4,1,8,2,7]=>0101010
[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]=>[5,6,3,4,2,7,1,8]=>0101010
[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[6,3,5,4,7,2,8,1]=>[3,6,4,5,2,7,1,8]=>0101010
[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[6,3,5,4,8,1,7,2]=>[3,6,4,5,1,8,2,7]=>0101010
[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[7,3,5,1,8,4,6,2]=>[2,6,4,8,1,5,3,7]=>0101010
[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[7,1,6,3,8,4,5,2]=>[2,8,3,6,1,5,4,7]=>0101010
[(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[7,1,6,5,8,3,4,2]=>[2,8,3,4,1,6,5,7]=>0101010
[(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[7,4,6,5,8,1,3,2]=>[2,5,3,4,1,8,6,7]=>0101010
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,8,4,6,3,7,5]=>[7,8,1,5,3,6,2,4]=>0101010
[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[8,1,6,3,4,2,7,5]=>[1,8,3,6,5,7,2,4]=>0101010
[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[8,3,5,2,6,1,7,4]=>[1,6,4,7,3,8,2,5]=>0101010
[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[8,2,6,3,5,1,7,4]=>[1,7,3,6,4,8,2,5]=>0101010
[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[8,2,7,1,5,3,6,4]=>[1,7,2,8,4,6,3,5]=>0101010
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,7,2,6,3,5,4]=>[1,8,2,7,3,6,4,5]=>0101010
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[8,1,7,2,4,3,6,5]=>[1,8,2,7,5,6,3,4]=>0101010
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[8,1,3,2,7,4,6,5]=>[1,8,6,7,2,5,3,4]=>0101010
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,7,4,6,5]=>[7,8,1,6,2,5,3,4]=>0101010
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,8,3,5,4,7,6]=>[7,8,1,6,4,5,2,3]=>0101010
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[8,1,3,2,5,4,7,6]=>[1,8,6,7,4,5,2,3]=>0101010
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[8,1,5,2,4,3,7,6]=>[1,8,4,7,5,6,2,3]=>0101010
[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,2,4,1,7,3,6,5]=>[1,7,5,8,2,6,3,4]=>0101010
[(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[3,2,8,4,6,1,7,5]=>[6,7,1,5,3,8,2,4]=>0101010
[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[3,2,8,1,7,4,6,5]=>[6,7,1,8,2,5,3,4]=>0101010
[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[3,2,8,1,5,4,7,6]=>[6,7,1,8,4,5,2,3]=>0101010
[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,2,4,1,8,5,7,6]=>[6,7,5,8,1,4,2,3]=>0101010
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>[5,8,6,7,1,4,2,3]=>0101010
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>[7,8,5,6,1,4,2,3]=>0101010
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5]=>[7,8,5,6,2,3,1,4]=>0101010
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,1,3,2,7,6,8,5]=>[5,8,6,7,2,3,1,4]=>0101010
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[6,7,5,8,2,3,1,4]=>0101010
[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,2,7,6,8,1,5,4]=>[6,7,2,3,1,8,4,5]=>0101010
[(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[3,2,7,1,8,5,6,4]=>[6,7,2,8,1,4,3,5]=>0101010
[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[3,2,6,5,8,1,7,4]=>[6,7,3,4,1,8,2,5]=>0101010
[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>[6,7,3,4,2,5,1,8]=>0101010
[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[6,5,7,2,4,3,8,1]=>[3,4,2,7,5,6,1,8]=>0101010
[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[7,2,4,1,8,5,6,3]=>[2,7,5,8,1,4,3,6]=>0101010
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[7,6,8,1,5,2,4,3]=>[2,3,1,8,4,7,5,6]=>0101010
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[7,6,8,1,3,2,5,4]=>[2,3,1,8,6,7,4,5]=>0101010
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,7,6,8,3,5,4]=>[7,8,2,3,1,6,4,5]=>0101010
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[7,1,8,5,6,2,4,3]=>[2,8,1,4,3,7,5,6]=>0101010
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[7,1,8,2,6,4,5,3]=>[2,8,1,7,3,5,4,6]=>0101010
[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[7,2,8,1,5,4,6,3]=>[2,7,1,8,4,5,3,6]=>0101010
[(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[6,2,8,4,5,1,7,3]=>[3,7,1,5,4,8,2,6]=>0101010
[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[5,4,8,2,6,1,7,3]=>[4,5,1,7,3,8,2,6]=>0101010
[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[6,1,8,4,7,2,5,3]=>[3,8,1,5,2,7,4,6]=>0101010
[(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[6,5,8,1,3,2,7,4]=>[3,4,1,8,6,7,2,5]=>0101010
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,6,5,8,3,7,4]=>[7,8,3,4,1,6,2,5]=>0101010
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[7,8,3,4,2,5,1,6]=>0101010
[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[6,5,7,4,8,1,3,2]=>[3,4,2,5,1,8,6,7]=>0101010
[(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[6,5,7,1,8,3,4,2]=>[3,4,2,8,1,6,5,7]=>0101010
[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>[4,5,3,6,2,7,1,8]=>0101010
[(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]=>[9,10,7,8,5,6,3,4,1,2]=>010101010
[(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]=>[11,12,9,10,7,8,5,6,3,4,1,2]=>01010101010
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the 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].
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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.
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.
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