Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>0=>1 [(1,2),(3,4)]=>[2,1,4,3]=>010=>101 [(1,3),(2,4)]=>[3,4,1,2]=>000=>001 [(1,4),(2,3)]=>[4,3,2,1]=>000=>001 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>01010=>10101 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>00010=>00101 [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>00010=>00101 [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>00000=>00001 [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>00000=>00001 [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>00000=>00001 [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>00000=>00001 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>00000=>00001 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>00000=>00001 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>01000=>10001 [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>01000=>10001 [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>00000=>00001 [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>00000=>00001 [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>00000=>00001 [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>00000=>00001 [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>0101010=>1010101 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>0001010=>0010101 [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>0001010=>0010101 [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>0000010=>0000101 [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>0000010=>0000101 [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>0000000=>0000001 [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>0000000=>0000001 [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>0000000=>0000001 [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>0000000=>0000001 [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>0000010=>0000101 [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>0000010=>0000101 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>0000010=>0000101 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>0000010=>0000101 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>0100010=>1000101 [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>0100010=>1000101 [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>0000010=>0000101 [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>0000010=>0000101 [(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>0000010=>0000101 [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>0000010=>0000101 [(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>0000000=>0000001 [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>0000000=>0000001 [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>0000000=>0000001 [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>0000000=>0000001 [(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>0000000=>0000001 [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>0000000=>0000001 [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>0000000=>0000001 [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>0000000=>0000001 [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>0100000=>1000001 [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>0100000=>1000001 [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>0000000=>0000001 [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>0000000=>0000001 [(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>0000000=>0000001 [(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>0000000=>0000001 [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>0000000=>0000001 [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>0000000=>0000001 [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>0000000=>0000001 [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>0000000=>0000001 [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>0000000=>0000001 [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>0000000=>0000001 [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>0000000=>0000001 [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>0000000=>0000001 [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>0100000=>1000001 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>0100000=>1000001 [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>0000000=>0000001 [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>0000000=>0000001 [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>0000000=>0000001 [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>0000000=>0000001 [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>0000000=>0000001 [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>0000000=>0000001 [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>0000000=>0000001 [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>0000000=>0000001 [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>0000000=>0000001 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>0000000=>0000001 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>0000000=>0000001 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>0000000=>0000001 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>0100000=>1000001 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>0100000=>1000001 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>0000000=>0000001 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>0000000=>0000001 [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>0000000=>0000001 [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>0000000=>0000001 [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>0000000=>0000001 [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>0000000=>0000001 [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>0000000=>0000001 [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>0000000=>0000001 [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>0000000=>0000001 [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>0000000=>0000001 [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>0001000=>0010001 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>0001000=>0010001 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>0101000=>1010001 [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>0101000=>1010001 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>0001000=>0010001 [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>0001000=>0010001 [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>0000000=>0000001 [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>0000000=>0000001 [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>0000000=>0000001 [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>0000000=>0000001 [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>0000000=>0000001 [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>0000000=>0000001 [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>0000000=>0000001 [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>0000000=>0000001 [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>0000000=>0000001 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>0000000=>0000001 [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>0100000=>1000001 [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>0100000=>1000001 [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>0000000=>0000001 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>0000000=>0000001 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>0000000=>0000001 [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>0000000=>0000001 [(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>0000000=>0000001 [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>0000000=>0000001 [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>0000000=>0000001 [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>0000000=>0000001 [(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>0000000=>0000001 [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>0000000=>0000001 [(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>0000000=>0000001 [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>0000000=>0000001 [(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>0100000=>1000001 [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>0100000=>1000001 [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>0000000=>0000001 [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>0000000=>0000001 [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>0000000=>0000001 [(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>0000000=>0000001 [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>0000000=>0000001 [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>0000000=>0000001
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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.
Map
valleys-to-peaks
Description
Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.