Identifier
Values
([],2) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([],3) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(1,2)],3) => [2,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 2
([],4) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 3
([(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => 3
([(1,3),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 2
([(0,3),(1,2)],4) => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 3
([(1,2),(1,3),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 2
([],5) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 4
([(3,4)],5) => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 4
([(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 3
([(1,4),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(1,4),(2,3)],5) => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => 2
([(1,4),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(0,1),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 3
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 3
([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(1,3),(1,4),(2,3),(2,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(2,5),(3,4)],6) => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 3
([(1,2),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 3
([(0,1),(2,5),(3,5),(4,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 3
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 3
([(0,1),(2,5),(3,4),(4,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3
([(1,6),(2,6),(3,5),(4,5)],7) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,3),(1,2),(4,6),(5,6)],7) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 3
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 3
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
([(0,7),(1,6),(2,5),(3,4)],8) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 3
([(0,2),(0,3),(1,2),(1,3),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(6,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(5,6),(5,7),(6,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,3),(1,2),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(6,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,7),(1,5),(2,4),(3,6),(4,5),(6,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,7),(1,6),(2,3),(2,6),(3,6),(4,5),(4,7),(5,7)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 3
search for individual values
searching the database for the individual values of this statistic
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
Map
swap returns and last descent
Description
Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map $\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.