Identifier
Values
[1,0] => [2,1] => [1,2] => [1,2] => 1
[1,0,1,0] => [3,1,2] => [1,3,2] => [3,1,2] => 1
[1,1,0,0] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [1,3,2,4] => [3,1,2,4] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [2,1,4,3] => [2,4,1,3] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [1,3,4,2] => [3,4,1,2] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,4,3,5,2] => [4,3,5,1,2] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,2,5,4] => [3,5,1,2,4] => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,1,5,4,3] => [5,2,4,1,3] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,1,4,3,5] => [2,4,1,3,5] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => 2
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,4,5,3,2] => [4,5,3,1,2] => 2
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,3,4,2,5] => [3,4,1,2,5] => 3
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [3,2,1,5,4] => [3,2,5,1,4] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,1,4,5,3] => [2,4,5,1,3] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,5,1,4,2] => [3,5,1,4,2] => 3
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,1,2] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,5,4,6,3,2] => [5,4,6,3,1,2] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,3,6,5,2] => [4,6,3,5,1,2] => 3
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,3,5,4,2] => [6,3,5,4,1,2] => 2
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [5,1,4,3,6,2] => [5,4,1,3,6,2] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,2,6,5,4] => [6,3,5,1,2,4] => 3
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 4
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,6,5,2,4,3] => [6,1,5,4,2,3] => 2
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,6,2,5,4,3] => [6,5,1,4,2,3] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,5,2,4,3,6] => [5,1,4,2,3,6] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [4,1,3,2,6,5] => [4,1,3,6,2,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,3,2,5,6,4] => [3,5,6,1,2,4] => 4
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,6,4,2,5,3] => [1,4,6,5,2,3] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [5,4,1,3,2,6] => [5,1,4,3,2,6] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,1,6,5,4,3] => [6,5,2,4,1,3] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,1,5,4,6,3] => [5,2,4,6,1,3] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,1,4,3,6,5] => [2,4,6,1,3,5] => 4
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,1,6,3,5,4] => [2,6,5,1,3,4] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [5,2,1,4,3,6] => [2,5,1,4,3,6] => 3
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,3,6,5,4,2] => [6,5,3,4,1,2] => 2
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,3,5,4,6,2] => [5,3,4,6,1,2] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,4,6,5,3,2] => [6,4,5,3,1,2] => 2
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,1,6,4,3,2] => [5,6,4,1,3,2] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,4,5,3,6,2] => [4,5,3,6,1,2] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,3,4,2,6,5] => [3,4,6,1,2,5] => 4
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => 4
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [5,1,3,4,2,6] => [1,5,3,4,2,6] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [3,2,1,6,5,4] => [3,6,2,5,1,4] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [3,2,1,5,4,6] => [3,2,5,1,4,6] => 3
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,1,4,6,5,3] => [6,2,4,5,1,3] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,1,5,6,4,3] => [5,6,2,4,1,3] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [2,1,4,5,3,6] => [2,4,5,1,3,6] => 4
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,6,1,5,4,2] => [6,3,5,1,4,2] => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,4,5,6,3,2] => [4,5,6,3,1,2] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [3,5,1,4,2,6] => [3,5,1,4,2,6] => 4
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [4,3,2,1,6,5] => [4,3,2,6,1,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [3,2,1,5,6,4] => [3,2,5,6,1,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [4,6,2,1,5,3] => [4,2,6,1,5,3] => 4
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,6,4,1,5,2] => [3,1,6,4,5,2] => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [1,3,2,5,6,4,7] => [3,5,6,1,2,4,7] => 5
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [1,3,7,6,5,4,2] => [7,6,5,3,4,1,2] => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => [1,3,6,5,7,4,2] => [6,5,7,3,4,1,2] => 3
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 1
[] => [1] => [1] => [1] => 0
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Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
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
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.