Identifier
Values
[1,0] => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,0,1,0] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,1,0,0] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9) => 0
[1,0,1,1,0,0] => [3,1,4,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 0
[1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 0
[1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9) => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9) => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 0
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,1,6,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [4,1,6,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,1,3,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,6,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 0
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,6,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,6,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 0
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,6,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [3,6,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [4,1,7,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [4,1,6,3,7,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [5,1,7,3,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [5,1,3,7,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [5,1,3,7,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [4,1,7,5,3,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => [3,5,2,7,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [4,7,2,5,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => [4,2,7,3,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => [3,7,5,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,3,4,1,7,2,6] => [3,7,4,2,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [4,2,7,5,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[] => [1] => [1] => ([(0,1)],2) => 0
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Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Map
lattice of intervals
Description
The lattice of intervals of a permutation.
An interval of a permutation $\pi$ is a possibly empty interval of values that appear in consecutive positions of $\pi$. The lattice of intervals of $\pi$ has as elements the intervals of $\pi$, ordered by set inclusion.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.