Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤ
Values
{{1},{2},{3}} => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
{{1,2,3,4}} => [2,3,4,1] => ([(1,2),(2,3)],4) => ([(0,2),(2,1)],3) => 1
{{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}} => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3,4,5}} => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => 1
{{1,2,3},{4,5}} => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => ([(0,2),(2,1)],3) => 1
{{1,2,4},{3,5}} => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => ([(0,2),(2,1)],3) => 1
{{1,3,5},{2,4}} => [3,4,5,2,1] => ([(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => 1
{{1,3},{2,4,5}} => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1},{2,3,4,5}} => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => ([(0,2),(2,1)],3) => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1},{2,4},{3,5}} => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => ([(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => ([(0,2),(2,1)],3) => 1
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,2),(2,1)],3) => 1
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => ([(0,2),(2,1)],3) => 1
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6) => ([(0,2),(2,1)],3) => 1
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
{{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2},{3,4,5},{6}} => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => ([(0,2),(2,1)],3) => 1
{{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
{{1,2,5},{3,6},{4}} => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 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
{{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => ([(0,2),(2,1)],3) => 1
{{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6) => ([(0,2),(2,1)],3) => 1
{{1,3,4,5},{2,6}} => [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1,3,4},{2,6},{5}} => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => ([(0,2),(2,1)],3) => 1
{{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
{{1,3,5},{2,4},{6}} => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
{{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
{{1,3},{2,4,5},{6}} => [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
{{1,3,6},{2,4},{5}} => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6) => ([(0,3),(2,1),(3,2)],4) => 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
{{1,3},{2,4},{5},{6}} => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],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
{{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6) => ([(0,2),(2,1)],3) => 1
{{1,4,5},{2,3,6}} => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1,4},{2,3,6},{5}} => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => ([(0,2),(2,1)],3) => 1
{{1,5},{2,3,4,6}} => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1,6},{2,3,4,5}} => [6,3,4,5,2,1] => ([(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
{{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => ([(0,3),(2,1),(3,2)],4) => 1
{{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,2),(2,1)],3) => 1
{{1},{2,3,4},{5,6}} => [1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => ([(0,2),(2,1)],3) => 1
{{1,5},{2,3,6},{4}} => [5,3,6,4,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
{{1},{2,3,5},{4,6}} => [1,3,5,6,2,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
{{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => ([(0,2),(2,1)],3) => 1
{{1,4,5},{2,6},{3}} => [4,6,3,5,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
{{1,4,5},{2},{3,6}} => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,2),(2,1)],3) => 1
{{1,4},{2,5,6},{3}} => [4,5,3,1,6,2] => ([(0,5),(1,4),(2,3),(2,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1,4},{2,5},{3},{6}} => [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1,4,6},{2},{3,5}} => [4,2,5,6,3,1] => ([(1,5),(2,3),(2,5),(5,4)],6) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => ([(1,4),(1,5),(2,3),(3,4),(3,5)],6) => ([(0,2),(2,1)],3) => 1
{{1,4},{2,6},{3},{5}} => [4,6,3,1,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,5)],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},{2,4,6},{3,5}} => [1,4,5,6,3,2] => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6) => ([(0,2),(2,1)],3) => 1
{{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
{{1},{2,4},{3,5},{6}} => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{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
{{1,5},{2},{3,4,6}} => [5,2,4,6,1,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,2),(2,1)],3) => 1
{{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => ([(0,2),(2,1)],3) => 1
{{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
{{1,5},{2},{3,6},{4}} => [5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
{{1,5},{2},{3},{4},{6}} => [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
{{1},{2,5},{3,6},{4}} => [1,5,6,4,2,3] => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1},{2},{3,5},{4,6}} => [1,2,5,6,3,4] => ([(0,5),(3,2),(4,1),(5,3),(5,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6) => ([(0,3),(2,1),(3,2)],4) => 1
{{1},{2,6},{3},{4},{5}} => [1,6,3,4,5,2] => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6) => ([(0,2),(2,1)],3) => 1
{{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
{{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
{{1,2,3,4,5,6},{7}} => [2,3,4,5,6,1,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
{{1,2,3,4,5},{6,7}} => [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
{{1,2,3,4,5},{6},{7}} => [2,3,4,5,1,6,7] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3,4,6},{5,7}} => [2,3,4,6,7,1,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 1
{{1,2,3,4},{5,6,7}} => [2,3,4,1,6,7,5] => ([(0,5),(0,6),(1,4),(3,5),(3,6),(4,3),(6,2)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
{{1,2,3,4},{5,6},{7}} => [2,3,4,1,6,5,7] => ([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3,4},{5},{6,7}} => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3,4},{5},{6},{7}} => [2,3,4,1,5,6,7] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => ([(0,2),(2,1)],3) => 1
{{1,2,3,5,7},{4,6}} => [2,3,5,6,7,4,1] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7) => ([(0,2),(2,1)],3) => 1
{{1,2,3,5},{4,6,7}} => [2,3,5,6,1,7,4] => ([(0,5),(0,6),(1,4),(2,6),(3,2),(4,3),(4,5)],7) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 1
{{1,2,3,5},{4,6},{7}} => [2,3,5,6,1,4,7] => ([(0,6),(1,4),(2,5),(3,2),(4,3),(4,6),(6,5)],7) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
{{1,2,3},{4,5,6,7}} => [2,3,1,5,6,7,4] => ([(0,5),(0,6),(1,3),(3,5),(3,6),(4,2),(6,4)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
{{1,2,3},{4,5,6},{7}} => [2,3,1,5,6,4,7] => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3},{4,5},{6,7}} => [2,3,1,5,4,7,6] => ([(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3},{4,5},{6},{7}} => [2,3,1,5,4,6,7] => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7) => ([(0,2),(2,1)],3) => 1
{{1,2,3,6},{4,7},{5}} => [2,3,6,7,5,1,4] => ([(0,6),(1,5),(4,3),(5,2),(5,4),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
{{1,2,3},{4,6},{5,7}} => [2,3,1,6,7,4,5] => ([(0,5),(0,6),(1,2),(2,5),(2,6),(5,4),(6,3)],7) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
{{1,2,3},{4},{5,6,7}} => [2,3,1,4,6,7,5] => ([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7) => ([(0,3),(2,1),(3,2)],4) => 1
{{1,2,3},{4},{5,6},{7}} => [2,3,1,4,6,5,7] => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7) => ([(0,2),(2,1)],3) => 1
{{1,2,3,7},{4},{5},{6}} => [2,3,7,4,5,6,1] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7) => ([(0,2),(2,1)],3) => 1
{{1,2,3},{4},{5},{6,7}} => [2,3,1,4,5,7,6] => ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7) => ([(0,2),(2,1)],3) => 1
{{1,2,4,5,6},{3,7}} => [2,4,7,5,6,1,3] => ([(0,6),(1,5),(1,6),(4,3),(5,2),(5,4)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
>>> Load all 501 entries. <<<
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.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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