Identifier
-
Mp00275:
Graphs
—to edge-partition of connected components⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001012: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1,0] => 2
([(1,2)],3) => [1] => [1,0] => 2
([(0,2),(1,2)],3) => [2] => [1,0,1,0] => 3
([(0,1),(0,2),(1,2)],3) => [3] => [1,0,1,0,1,0] => 3
([(2,3)],4) => [1] => [1,0] => 2
([(1,3),(2,3)],4) => [2] => [1,0,1,0] => 3
([(0,3),(1,3),(2,3)],4) => [3] => [1,0,1,0,1,0] => 3
([(0,3),(1,2)],4) => [1,1] => [1,1,0,0] => 3
([(0,3),(1,2),(2,3)],4) => [3] => [1,0,1,0,1,0] => 3
([(1,2),(1,3),(2,3)],4) => [3] => [1,0,1,0,1,0] => 3
([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(3,4)],5) => [1] => [1,0] => 2
([(2,4),(3,4)],5) => [2] => [1,0,1,0] => 3
([(1,4),(2,4),(3,4)],5) => [3] => [1,0,1,0,1,0] => 3
([(0,4),(1,4),(2,4),(3,4)],5) => [4] => [1,0,1,0,1,0,1,0] => 3
([(1,4),(2,3)],5) => [1,1] => [1,1,0,0] => 3
([(1,4),(2,3),(3,4)],5) => [3] => [1,0,1,0,1,0] => 3
([(0,1),(2,4),(3,4)],5) => [2,1] => [1,0,1,1,0,0] => 4
([(2,3),(2,4),(3,4)],5) => [3] => [1,0,1,0,1,0] => 3
([(0,4),(1,4),(2,3),(3,4)],5) => [4] => [1,0,1,0,1,0,1,0] => 3
([(1,4),(2,3),(2,4),(3,4)],5) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,4),(1,3),(2,3),(2,4)],5) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,0,1,0,1,1,0,0] => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(4,5)],6) => [1] => [1,0] => 2
([(3,5),(4,5)],6) => [2] => [1,0,1,0] => 3
([(2,5),(3,5),(4,5)],6) => [3] => [1,0,1,0,1,0] => 3
([(1,5),(2,5),(3,5),(4,5)],6) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(2,5),(3,4)],6) => [1,1] => [1,1,0,0] => 3
([(2,5),(3,4),(4,5)],6) => [3] => [1,0,1,0,1,0] => 3
([(1,2),(3,5),(4,5)],6) => [2,1] => [1,0,1,1,0,0] => 4
([(3,4),(3,5),(4,5)],6) => [3] => [1,0,1,0,1,0] => 3
([(1,5),(2,5),(3,4),(4,5)],6) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => 4
([(2,5),(3,4),(3,5),(4,5)],6) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [1,1,1,0,0,0] => 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,1,0,1,0,0] => 4
([(1,5),(2,4),(3,4),(3,5)],6) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => 4
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 4
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [1,0,1,1,1,0,0,0] => 5
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
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search for individual values
searching the database for the individual values of this statistic
Description
Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.
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.
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 edge-partition of connected components
Description
Sends a graph to the partition recording the number of edges in its connected components.
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