Identifier
-
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000105: Set partitions ⟶ ℤ
Values
[(1,2)] => [2,1] => [1,2] => {{1},{2}} => 2
[(1,2),(3,4)] => [2,1,4,3] => [1,4,3,2] => {{1},{2,4},{3}} => 3
[(1,3),(2,4)] => [3,4,1,2] => [4,1,2,3] => {{1,2,3,4}} => 1
[(1,4),(2,3)] => [4,3,2,1] => [3,2,1,4] => {{1,3},{2},{4}} => 3
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,4,3,6,5,2] => {{1},{2,4,6},{3},{5}} => 4
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,1,2,6,5,3] => {{1,2,3,4,6},{5}} => 2
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [3,2,1,6,5,4] => {{1,3},{2},{4,6},{5}} => 4
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [3,2,6,1,4,5] => {{1,3,4,5,6},{2}} => 2
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,2,5,4,1,6] => {{1,3,5},{2},{4},{6}} => 4
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,5,2,3,1,6] => {{1,2,3,4,5},{6}} => 2
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [4,6,2,1,3,5] => {{1,4},{2,3,5,6}} => 2
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [5,6,1,2,3,4] => {{1,3,5},{2,4,6}} => 2
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [5,1,6,2,4,3] => {{1,2,4,5},{3,6}} => 2
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [1,5,6,3,4,2] => {{1},{2,3,4,5,6}} => 2
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [1,6,5,4,3,2] => {{1},{2,6},{3,5},{4}} => 4
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => {{1,2,3,5,6},{4}} => 2
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => {{1,3,4,6},{2,5}} => 2
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [6,4,3,1,2,5] => {{1,2,4,5,6},{3}} => 2
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [5,4,3,2,1,6] => {{1,5},{2,4},{3},{6}} => 4
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Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous Stirling numbers of the second kind $S_2(n,k)$ given by the number of set partitions of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
The generating function of this statistic yields the famous Stirling numbers of the second kind $S_2(n,k)$ given by the number of set partitions of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
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