Identifier
Values
[(1,3),(2,4)] => [3,4,1,2] => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) => ([(0,2),(2,1)],3) => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 2
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => ([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8) => ([(0,3),(2,1),(3,2)],4) => 1
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => ([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => ([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8) => ([(0,2),(2,1)],3) => 1
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => ([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => ([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(5,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6)],8) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => ([(4,7),(5,6)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => ([(0,5),(0,7),(1,4),(1,6),(6,2),(7,3)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => ([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(4,7),(5,6)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(6,3),(7,2)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,3),(7,2)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,6),(5,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => ([(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 2
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => ([(2,6),(2,7),(3,6),(3,7),(6,5),(7,4)],8) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => ([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => ([(0,7),(1,6),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => ([(0,5),(1,4),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[(1,6),(2,5),(3,8),(4,7)] => [6,5,8,7,2,1,4,3] => ([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,8),(2,6),(3,7),(4,5)] => [8,6,7,5,4,2,3,1] => ([(4,7),(5,6)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,5),(2,7),(3,8),(4,6)] => [5,7,8,6,1,4,2,3] => ([(0,5),(0,7),(1,4),(1,6),(6,2),(7,3)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,4),(2,7),(3,8),(5,6)] => [4,7,8,1,6,5,2,3] => ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(4,2),(5,3)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,3),(2,7),(4,8),(5,6)] => [3,7,1,8,6,5,2,4] => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,5),(3,4)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,2),(3,7),(4,8),(5,6)] => [2,1,7,8,6,5,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,3),(7,2)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[(1,7),(2,8),(3,6),(4,5)] => [7,8,6,5,4,3,1,2] => ([(4,7),(5,6)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
search for individual values
searching the database for the individual values of this statistic
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Map
antichains of maximal size
Description
The lattice of antichains of maximal size in a poset.
The set of antichains of maximal size can be ordered by setting $A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B$, where $\mathop{\downarrow} A$ is the order ideal generated by $A$.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.