Identifier
-
Mp00276:
Graphs
—to edge-partition of biconnected components⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001183: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(1,2)],3) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(0,2),(1,2)],3) => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => 3
([(2,3)],4) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(1,3),(2,3)],4) => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => 3
([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(3,4)],5) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(2,4),(3,4)],5) => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => 3
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 4
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 4
([(4,5)],6) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(3,5),(4,5)],6) => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 6
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => 3
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 6
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 4
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 6
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 6
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 6
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 4
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 4
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 4
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 6
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 4
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 4
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 4
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 4
([(5,6)],7) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(4,6),(5,6)],7) => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 4
([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
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Description
The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
Map
to edge-partition of biconnected components
Description
Sends a graph to the partition recording the number of edges in its biconnected components.
The biconnected components are also known as blocks of a graph.
The biconnected components are also known as blocks of a graph.
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