Identifier
Mp00092:
Perfect matchings
—to set partition⟶
Set partitions
Mp00219: Set partitions —inverse Yip⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00219: Set partitions —inverse Yip⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
[(1,2)] => {{1,2}} => {{1,2}} => [2,1] => 0
[(1,2),(3,4)] => {{1,2},{3,4}} => {{1,2,4},{3}} => [2,4,3,1] => 000
[(1,3),(2,4)] => {{1,3},{2,4}} => {{1,4},{2,3}} => [4,3,2,1] => 000
[(1,4),(2,3)] => {{1,4},{2,3}} => {{1,3},{2,4}} => [3,4,1,2] => 000
[(1,2),(3,4),(5,6)] => {{1,2},{3,4},{5,6}} => {{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => 00000
[(1,3),(2,4),(5,6)] => {{1,3},{2,4},{5,6}} => {{1,4},{2,3,6},{5}} => [4,3,6,1,5,2] => 00000
[(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => {{1,3,6},{2,4},{5}} => [3,4,6,2,5,1] => 00000
[(1,5),(2,3),(4,6)] => {{1,5},{2,3},{4,6}} => {{1,3,5},{2,6},{4}} => [3,6,5,4,1,2] => 00000
[(1,6),(2,3),(4,5)] => {{1,6},{2,3},{4,5}} => {{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => 00000
[(1,6),(2,4),(3,5)] => {{1,6},{2,4},{3,5}} => {{1,5},{2,4},{3,6}} => [5,4,6,2,1,3] => 00000
[(1,5),(2,4),(3,6)] => {{1,5},{2,4},{3,6}} => {{1,6},{2,4},{3,5}} => [6,4,5,2,3,1] => 00000
[(1,4),(2,5),(3,6)] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => 00000
[(1,3),(2,5),(4,6)] => {{1,3},{2,5},{4,6}} => {{1,6},{2,3,5},{4}} => [6,3,5,4,2,1] => 00000
[(1,2),(3,5),(4,6)] => {{1,2},{3,5},{4,6}} => {{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => 00000
[(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => {{1,2,5},{3,6},{4}} => [2,5,6,4,1,3] => 00000
[(1,3),(2,6),(4,5)] => {{1,3},{2,6},{4,5}} => {{1,5},{2,3,6},{4}} => [5,3,6,4,1,2] => 00000
[(1,4),(2,6),(3,5)] => {{1,4},{2,6},{3,5}} => {{1,5},{2,6},{3,4}} => [5,6,4,3,1,2] => 00000
[(1,5),(2,6),(3,4)] => {{1,5},{2,6},{3,4}} => {{1,4},{2,6},{3,5}} => [4,6,5,1,3,2] => 00000
[(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => {{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => 00000
[(1,2),(3,4),(5,6),(7,8)] => {{1,2},{3,4},{5,6},{7,8}} => {{1,2,4,8},{3,6},{5},{7}} => [2,4,6,8,5,3,7,1] => 0000000
[(1,3),(2,4),(5,6),(7,8)] => {{1,3},{2,4},{5,6},{7,8}} => {{1,4,8},{2,3,6},{5},{7}} => [4,3,6,8,5,2,7,1] => 0000000
[(1,4),(2,3),(5,6),(7,8)] => {{1,4},{2,3},{5,6},{7,8}} => {{1,3,6},{2,4,8},{5},{7}} => [3,4,6,8,5,1,7,2] => 0000000
[(1,5),(2,3),(4,6),(7,8)] => {{1,5},{2,3},{4,6},{7,8}} => {{1,3,5},{2,6},{4,8},{7}} => [3,6,5,8,1,2,7,4] => 0000000
[(1,6),(2,3),(4,5),(7,8)] => {{1,6},{2,3},{4,5},{7,8}} => {{1,3,6},{2,5},{4,8},{7}} => [3,5,6,8,2,1,7,4] => 0000000
[(1,7),(2,3),(4,5),(6,8)] => {{1,7},{2,3},{4,5},{6,8}} => {{1,3,8},{2,5},{4,7},{6}} => [3,5,8,7,2,6,4,1] => 0000000
[(1,8),(2,3),(4,5),(6,7)] => {{1,8},{2,3},{4,5},{6,7}} => {{1,3,7},{2,5},{4,8},{6}} => [3,5,7,8,2,6,1,4] => 0000000
[(1,8),(2,4),(3,5),(6,7)] => {{1,8},{2,4},{3,5},{6,7}} => {{1,5},{2,4,8},{3,7},{6}} => [5,4,7,8,1,6,3,2] => 0000000
[(1,7),(2,4),(3,5),(6,8)] => {{1,7},{2,4},{3,5},{6,8}} => {{1,5},{2,4,7},{3,8},{6}} => [5,4,8,7,1,6,2,3] => 0000000
[(1,6),(2,4),(3,5),(7,8)] => {{1,6},{2,4},{3,5},{7,8}} => {{1,5},{2,4,8},{3,6},{7}} => [5,4,6,8,1,3,7,2] => 0000000
[(1,5),(2,4),(3,6),(7,8)] => {{1,5},{2,4},{3,6},{7,8}} => {{1,6},{2,4,8},{3,5},{7}} => [6,4,5,8,3,1,7,2] => 0000000
[(1,4),(2,5),(3,6),(7,8)] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4,8},{7}} => [6,5,4,8,2,1,7,3] => 0000000
[(1,3),(2,5),(4,6),(7,8)] => {{1,3},{2,5},{4,6},{7,8}} => {{1,6},{2,3,5},{4,8},{7}} => [6,3,5,8,2,1,7,4] => 0000000
[(1,2),(3,5),(4,6),(7,8)] => {{1,2},{3,5},{4,6},{7,8}} => {{1,2,6},{3,5},{4,8},{7}} => [2,6,5,8,3,1,7,4] => 0000000
[(1,2),(3,6),(4,5),(7,8)] => {{1,2},{3,6},{4,5},{7,8}} => {{1,2,5},{3,6},{4,8},{7}} => [2,5,6,8,1,3,7,4] => 0000000
[(1,3),(2,6),(4,5),(7,8)] => {{1,3},{2,6},{4,5},{7,8}} => {{1,5},{2,3,6},{4,8},{7}} => [5,3,6,8,1,2,7,4] => 0000000
[(1,4),(2,6),(3,5),(7,8)] => {{1,4},{2,6},{3,5},{7,8}} => {{1,5},{2,6},{3,4,8},{7}} => [5,6,4,8,1,2,7,3] => 0000000
[(1,5),(2,6),(3,4),(7,8)] => {{1,5},{2,6},{3,4},{7,8}} => {{1,4,8},{2,6},{3,5},{7}} => [4,6,5,8,3,2,7,1] => 0000000
[(1,6),(2,5),(3,4),(7,8)] => {{1,6},{2,5},{3,4},{7,8}} => {{1,4,8},{2,5},{3,6},{7}} => [4,5,6,8,2,3,7,1] => 0000000
[(1,7),(2,5),(3,4),(6,8)] => {{1,7},{2,5},{3,4},{6,8}} => {{1,4,7},{2,5},{3,8},{6}} => [4,5,8,7,2,6,1,3] => 0000000
[(1,8),(2,5),(3,4),(6,7)] => {{1,8},{2,5},{3,4},{6,7}} => {{1,4,8},{2,5},{3,7},{6}} => [4,5,7,8,2,6,3,1] => 0000000
[(1,8),(2,6),(3,4),(5,7)] => {{1,8},{2,6},{3,4},{5,7}} => {{1,4,8},{2,7},{3,6},{5}} => [4,7,6,8,5,3,2,1] => 0000000
[(1,7),(2,6),(3,4),(5,8)] => {{1,7},{2,6},{3,4},{5,8}} => {{1,4,7},{2,8},{3,6},{5}} => [4,8,6,7,5,3,1,2] => 0000000
[(1,6),(2,7),(3,4),(5,8)] => {{1,6},{2,7},{3,4},{5,8}} => {{1,4,6},{2,8},{3,7},{5}} => [4,8,7,6,5,1,3,2] => 0000000
[(1,5),(2,7),(3,4),(6,8)] => {{1,5},{2,7},{3,4},{6,8}} => {{1,4,7},{2,8},{3,5},{6}} => [4,8,5,7,3,6,1,2] => 0000000
[(1,4),(2,7),(3,5),(6,8)] => {{1,4},{2,7},{3,5},{6,8}} => {{1,5},{2,8},{3,4,7},{6}} => [5,8,4,7,1,6,3,2] => 0000000
[(1,3),(2,7),(4,5),(6,8)] => {{1,3},{2,7},{4,5},{6,8}} => {{1,5},{2,3,8},{4,7},{6}} => [5,3,8,7,1,6,4,2] => 0000000
[(1,2),(3,7),(4,5),(6,8)] => {{1,2},{3,7},{4,5},{6,8}} => {{1,2,5},{3,8},{4,7},{6}} => [2,5,8,7,1,6,4,3] => 0000000
[(1,2),(3,8),(4,5),(6,7)] => {{1,2},{3,8},{4,5},{6,7}} => {{1,2,5},{3,7},{4,8},{6}} => [2,5,7,8,1,6,3,4] => 0000000
[(1,3),(2,8),(4,5),(6,7)] => {{1,3},{2,8},{4,5},{6,7}} => {{1,5},{2,3,7},{4,8},{6}} => [5,3,7,8,1,6,2,4] => 0000000
[(1,4),(2,8),(3,5),(6,7)] => {{1,4},{2,8},{3,5},{6,7}} => {{1,5},{2,7},{3,4,8},{6}} => [5,7,4,8,1,6,2,3] => 0000000
[(1,5),(2,8),(3,4),(6,7)] => {{1,5},{2,8},{3,4},{6,7}} => {{1,4,8},{2,7},{3,5},{6}} => [4,7,5,8,3,6,2,1] => 0000000
[(1,6),(2,8),(3,4),(5,7)] => {{1,6},{2,8},{3,4},{5,7}} => {{1,4,6},{2,7},{3,8},{5}} => [4,7,8,6,5,1,2,3] => 0000000
[(1,7),(2,8),(3,4),(5,6)] => {{1,7},{2,8},{3,4},{5,6}} => {{1,4,7},{2,6},{3,8},{5}} => [4,6,8,7,5,2,1,3] => 0000000
[(1,8),(2,7),(3,4),(5,6)] => {{1,8},{2,7},{3,4},{5,6}} => {{1,4,8},{2,6},{3,7},{5}} => [4,6,7,8,5,2,3,1] => 0000000
[(1,8),(2,7),(3,5),(4,6)] => {{1,8},{2,7},{3,5},{4,6}} => {{1,6},{2,5},{3,7},{4,8}} => [6,5,7,8,2,1,3,4] => 0000000
[(1,7),(2,8),(3,5),(4,6)] => {{1,7},{2,8},{3,5},{4,6}} => {{1,6},{2,5},{3,8},{4,7}} => [6,5,8,7,2,1,4,3] => 0000000
[(1,6),(2,8),(3,5),(4,7)] => {{1,6},{2,8},{3,5},{4,7}} => {{1,7},{2,5},{3,8},{4,6}} => [7,5,8,6,2,4,1,3] => 0000000
[(1,5),(2,8),(3,6),(4,7)] => {{1,5},{2,8},{3,6},{4,7}} => {{1,7},{2,6},{3,8},{4,5}} => [7,6,8,5,4,2,1,3] => 0000000
[(1,4),(2,8),(3,6),(5,7)] => {{1,4},{2,8},{3,6},{5,7}} => {{1,7},{2,6},{3,4,8},{5}} => [7,6,4,8,5,2,1,3] => 0000000
[(1,3),(2,8),(4,6),(5,7)] => {{1,3},{2,8},{4,6},{5,7}} => {{1,7},{2,3,6},{4,8},{5}} => [7,3,6,8,5,2,1,4] => 0000000
[(1,2),(3,8),(4,6),(5,7)] => {{1,2},{3,8},{4,6},{5,7}} => {{1,2,7},{3,6},{4,8},{5}} => [2,7,6,8,5,3,1,4] => 0000000
[(1,2),(3,7),(4,6),(5,8)] => {{1,2},{3,7},{4,6},{5,8}} => {{1,2,8},{3,6},{4,7},{5}} => [2,8,6,7,5,3,4,1] => 0000000
[(1,3),(2,7),(4,6),(5,8)] => {{1,3},{2,7},{4,6},{5,8}} => {{1,8},{2,3,6},{4,7},{5}} => [8,3,6,7,5,2,4,1] => 0000000
[(1,4),(2,7),(3,6),(5,8)] => {{1,4},{2,7},{3,6},{5,8}} => {{1,8},{2,6},{3,4,7},{5}} => [8,6,4,7,5,2,3,1] => 0000000
[(1,5),(2,7),(3,6),(4,8)] => {{1,5},{2,7},{3,6},{4,8}} => {{1,8},{2,6},{3,7},{4,5}} => [8,6,7,5,4,2,3,1] => 0000000
[(1,6),(2,7),(3,5),(4,8)] => {{1,6},{2,7},{3,5},{4,8}} => {{1,8},{2,5},{3,7},{4,6}} => [8,5,7,6,2,4,3,1] => 0000000
[(1,7),(2,6),(3,5),(4,8)] => {{1,7},{2,6},{3,5},{4,8}} => {{1,8},{2,5},{3,6},{4,7}} => [8,5,6,7,2,3,4,1] => 0000000
[(1,8),(2,6),(3,5),(4,7)] => {{1,8},{2,6},{3,5},{4,7}} => {{1,7},{2,5},{3,6},{4,8}} => [7,5,6,8,2,3,1,4] => 0000000
[(1,8),(2,5),(3,6),(4,7)] => {{1,8},{2,5},{3,6},{4,7}} => {{1,7},{2,6},{3,5},{4,8}} => [7,6,5,8,3,2,1,4] => 0000000
[(1,7),(2,5),(3,6),(4,8)] => {{1,7},{2,5},{3,6},{4,8}} => {{1,8},{2,6},{3,5},{4,7}} => [8,6,5,7,3,2,4,1] => 0000000
[(1,6),(2,5),(3,7),(4,8)] => {{1,6},{2,5},{3,7},{4,8}} => {{1,8},{2,7},{3,5},{4,6}} => [8,7,5,6,3,4,2,1] => 0000000
[(1,5),(2,6),(3,7),(4,8)] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => [8,7,6,5,4,3,2,1] => 0000000
[(1,4),(2,6),(3,7),(5,8)] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4,6},{5}} => [8,7,4,6,5,3,2,1] => 0000000
[(1,3),(2,6),(4,7),(5,8)] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3,7},{4,6},{5}} => [8,3,7,6,5,4,2,1] => 0000000
[(1,2),(3,6),(4,7),(5,8)] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2,8},{3,7},{4,6},{5}} => [2,8,7,6,5,4,3,1] => 0000000
[(1,2),(3,5),(4,7),(6,8)] => {{1,2},{3,5},{4,7},{6,8}} => {{1,2,8},{3,5},{4,7},{6}} => [2,8,5,7,3,6,4,1] => 0000000
[(1,3),(2,5),(4,7),(6,8)] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3,5},{4,7},{6}} => [8,3,5,7,2,6,4,1] => 0000000
[(1,4),(2,5),(3,7),(6,8)] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4,7},{6}} => [8,5,4,7,2,6,3,1] => 0000000
[(1,5),(2,4),(3,7),(6,8)] => {{1,5},{2,4},{3,7},{6,8}} => {{1,8},{2,4,7},{3,5},{6}} => [8,4,5,7,3,6,2,1] => 0000000
[(1,6),(2,4),(3,7),(5,8)] => {{1,6},{2,4},{3,7},{5,8}} => {{1,8},{2,4,6},{3,7},{5}} => [8,4,7,6,5,2,3,1] => 0000000
[(1,7),(2,4),(3,6),(5,8)] => {{1,7},{2,4},{3,6},{5,8}} => {{1,8},{2,4,7},{3,6},{5}} => [8,4,6,7,5,3,2,1] => 0000000
[(1,8),(2,4),(3,6),(5,7)] => {{1,8},{2,4},{3,6},{5,7}} => {{1,7},{2,4,8},{3,6},{5}} => [7,4,6,8,5,3,1,2] => 0000000
[(1,8),(2,3),(4,6),(5,7)] => {{1,8},{2,3},{4,6},{5,7}} => {{1,3,6},{2,7},{4,8},{5}} => [3,7,6,8,5,1,2,4] => 0000000
[(1,7),(2,3),(4,6),(5,8)] => {{1,7},{2,3},{4,6},{5,8}} => {{1,3,6},{2,8},{4,7},{5}} => [3,8,6,7,5,1,4,2] => 0000000
[(1,6),(2,3),(4,7),(5,8)] => {{1,6},{2,3},{4,7},{5,8}} => {{1,3,7},{2,8},{4,6},{5}} => [3,8,7,6,5,4,1,2] => 0000000
[(1,5),(2,3),(4,7),(6,8)] => {{1,5},{2,3},{4,7},{6,8}} => {{1,3,5},{2,8},{4,7},{6}} => [3,8,5,7,1,6,4,2] => 0000000
[(1,4),(2,3),(5,7),(6,8)] => {{1,4},{2,3},{5,7},{6,8}} => {{1,3,8},{2,4,7},{5},{6}} => [3,4,8,7,5,6,2,1] => 0000000
[(1,3),(2,4),(5,7),(6,8)] => {{1,3},{2,4},{5,7},{6,8}} => {{1,4,7},{2,3,8},{5},{6}} => [4,3,8,7,5,6,1,2] => 0000000
[(1,2),(3,4),(5,7),(6,8)] => {{1,2},{3,4},{5,7},{6,8}} => {{1,2,4,7},{3,8},{5},{6}} => [2,4,8,7,5,6,1,3] => 0000000
[(1,2),(3,4),(5,8),(6,7)] => {{1,2},{3,4},{5,8},{6,7}} => {{1,2,4,8},{3,7},{5},{6}} => [2,4,7,8,5,6,3,1] => 0000000
[(1,3),(2,4),(5,8),(6,7)] => {{1,3},{2,4},{5,8},{6,7}} => {{1,4,8},{2,3,7},{5},{6}} => [4,3,7,8,5,6,2,1] => 0000000
[(1,4),(2,3),(5,8),(6,7)] => {{1,4},{2,3},{5,8},{6,7}} => {{1,3,7},{2,4,8},{5},{6}} => [3,4,7,8,5,6,1,2] => 0000000
[(1,5),(2,3),(4,8),(6,7)] => {{1,5},{2,3},{4,8},{6,7}} => {{1,3,5},{2,7},{4,8},{6}} => [3,7,5,8,1,6,2,4] => 0000000
[(1,6),(2,3),(4,8),(5,7)] => {{1,6},{2,3},{4,8},{5,7}} => {{1,3,8},{2,7},{4,6},{5}} => [3,7,8,6,5,4,2,1] => 0000000
[(1,7),(2,3),(4,8),(5,6)] => {{1,7},{2,3},{4,8},{5,6}} => {{1,3,8},{2,6},{4,7},{5}} => [3,6,8,7,5,2,4,1] => 0000000
[(1,8),(2,3),(4,7),(5,6)] => {{1,8},{2,3},{4,7},{5,6}} => {{1,3,7},{2,6},{4,8},{5}} => [3,6,7,8,5,2,1,4] => 0000000
[(1,8),(2,4),(3,7),(5,6)] => {{1,8},{2,4},{3,7},{5,6}} => {{1,6},{2,4,8},{3,7},{5}} => [6,4,7,8,5,1,3,2] => 0000000
[(1,7),(2,4),(3,8),(5,6)] => {{1,7},{2,4},{3,8},{5,6}} => {{1,6},{2,4,7},{3,8},{5}} => [6,4,8,7,5,1,2,3] => 0000000
[(1,6),(2,4),(3,8),(5,7)] => {{1,6},{2,4},{3,8},{5,7}} => {{1,7},{2,4,6},{3,8},{5}} => [7,4,8,6,5,2,1,3] => 0000000
[(1,5),(2,4),(3,8),(6,7)] => {{1,5},{2,4},{3,8},{6,7}} => {{1,7},{2,4,8},{3,5},{6}} => [7,4,5,8,3,6,1,2] => 0000000
[(1,4),(2,5),(3,8),(6,7)] => {{1,4},{2,5},{3,8},{6,7}} => {{1,7},{2,5},{3,4,8},{6}} => [7,5,4,8,2,6,1,3] => 0000000
>>> Load all 126 entries. <<<Map
to set partition
Description
Return the set partition corresponding to the perfect matching.
Map
inverse Yip
Description
The inverse of a transformation of set partitions due to Yip.
Return the set partition of $\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Yip's bijection $\psi^{-1}$.
Return the set partition of $\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Yip's bijection $\psi^{-1}$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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.
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.
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