Identifier
Values
[[[]]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[[]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[[]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]]],[]] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[[[]]]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]],[[]]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[[]]],[]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[],[[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[[]],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[[[]]],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[]],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[]],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[[]]],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[],[[[]]]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[[[]]],[]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[[]],[],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[[],[[]]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[[[]],[]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[]],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]]],[[],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[[]]],[[]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]],[]],[[]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[[[]]]],[]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[[]]],[]],[]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[],[[[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[[]],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[[[]]],[]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]],[],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]],[[]],[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[[]]],[],[]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[],[[[]]]]]] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[[]],[[]]]]] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[[[]]],[]]]] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[],[],[[[]]]] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[],[[]],[[]]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[],[[[]]],[]] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[[]],[],[[]]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[[]],[[]],[]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[],[[[]]],[],[]] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[]],[],[],[[]]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[]],[],[[]],[]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[]],[[]],[],[]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[[]]],[],[],[]] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[],[]],[[[]]]] => ([(0,7),(1,6),(2,6),(3,4),(4,5),(5,7),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[],[[[]]],[[],[]]] => ([(0,7),(1,6),(2,6),(3,4),(4,5),(5,7),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[],[],[],[[]]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[],[],[[]],[]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[],[[]],[],[]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[]],[[]],[],[],[]] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]]],[],[],[],[]] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[]],[],[[[]]]] => ([(0,7),(1,6),(2,6),(3,4),(4,5),(5,7),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]]],[],[[],[]]] => ([(0,7),(1,6),(2,6),(3,4),(4,5),(5,7),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[],[]],[[[]]],[]] => ([(0,7),(1,6),(2,6),(3,4),(4,5),(5,7),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[[[]]],[[],[]],[]] => ([(0,7),(1,6),(2,6),(3,4),(4,5),(5,7),(6,7)],8) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
search for individual values
searching the database for the individual values of this statistic
Description
Number of indecomposable injective modules with projective dimension 2.
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
lattice of congruences
Description
The lattice of congruences of a lattice.
A congruence of a lattice is an equivalence relation such that $a_1 \cong a_2$ and $b_1 \cong b_2$ implies $a_1 \vee b_1 \cong a_2 \vee b_2$ and $a_1 \wedge b_1 \cong a_2 \wedge b_2$.
The set of congruences ordered by refinement forms a lattice.
Map
to poset
Description
Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.