Identifier
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
=>
Cc0005;cc-rep-0Cc0012;cc-rep-1
[1,0]=>[(1,2)]=>[2,1]=>[2,1]=>0 [1,0,1,0]=>[(1,2),(3,4)]=>[2,1,4,3]=>[2,4,1,3]=>000 [1,1,0,0]=>[(1,4),(2,3)]=>[4,3,2,1]=>[4,3,2,1]=>000 [1,0,1,0,1,0]=>[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,4,6,1,3,5]=>00000 [1,0,1,1,0,0]=>[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[6,5,2,4,1,3]=>00000 [1,1,0,0,1,0]=>[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,3,2,6,1,5]=>00000 [1,1,0,1,0,0]=>[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,6,2,5,4,1]=>00000 [1,1,1,0,0,0]=>[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>00000 [1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,4,6,8,1,3,5,7]=>0000000 [1,0,1,0,1,1,0,0]=>[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[8,7,2,4,6,1,3,5]=>0000000 [1,0,1,1,0,0,1,0]=>[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[6,5,2,4,8,1,3,7]=>0000000 [1,0,1,1,0,1,0,0]=>[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,8,5,7,4,6,1,3]=>0000000 [1,0,1,1,1,0,0,0]=>[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[8,7,6,5,2,4,1,3]=>0000000 [1,1,0,0,1,0,1,0]=>[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[4,3,2,6,8,1,5,7]=>0000000 [1,1,0,0,1,1,0,0]=>[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[4,8,3,7,2,6,1,5]=>0000000 [1,1,0,1,0,0,1,0]=>[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,6,2,5,4,8,1,7]=>0000000 [1,1,0,1,0,1,0,0]=>[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,5,8,2,4,7,6,1]=>0000000 [1,1,0,1,1,0,0,0]=>[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[8,7,3,6,2,5,4,1]=>0000000 [1,1,1,0,0,0,1,0]=>[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,8,1,7]=>0000000 [1,1,1,0,0,1,0,0]=>[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[5,4,3,8,2,7,6,1]=>0000000 [1,1,1,0,1,0,0,0]=>[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,8,3,7,6,5,2,1]=>0000000 [1,1,1,1,0,0,0,0]=>[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000
Map
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
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.