Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Images
[(1,2)] => [2,1] => [1,2] => [2,1]
[(1,2),(3,4)] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3]
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3]
[(1,4),(2,3)] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => [2,1,4,3,6,5]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [5,6,3,1,4,2] => [2,1,4,6,3,5]
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [5,6,1,2,3,4] => [2,1,6,5,4,3]
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [2,3,5,1,6,4] => [5,4,2,6,1,3]
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [2,3,6,1,4,5] => [5,4,1,6,3,2]
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,6,2,5,1,3] => [3,1,5,2,6,4]
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [4,5,1,2,6,3] => [3,2,6,5,1,4]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [5,4,1,6,3,2] => [2,3,6,1,4,5]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [4,6,2,5,1,3]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [5,3,6,4,1,2] => [2,4,1,3,6,5]
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [3,4,5,6,1,2] => [4,3,2,1,6,5]
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,1,6,2,4,5] => [4,6,1,5,3,2]
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,1,5,6,2,3] => [3,6,2,1,5,4]
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [5,1,3,4,6,2] => [2,6,4,3,1,5]
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
weak order rowmotion
Description
Return the reversal of the permutation obtained by inverting the corresponding Laguerre heap.
This map is the composite of Mp00241invert Laguerre heap and Mp00064reverse.
Conjecturally, it is also the rowmotion on the weak order:
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$.
Note that the dynamics of weak order rowmotion is poorly understood. A collection of nontrivial homomesies is described in Corollary 6.14 of [4].
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$