Identifier
-
Mp00276:
Graphs
—to edge-partition of biconnected components⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001000: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1,0,1,0] => [1,0,1,0] => 1
([(1,2)],3) => [1] => [1,0,1,0] => [1,0,1,0] => 1
([(0,2),(1,2)],3) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(2,3)],4) => [1] => [1,0,1,0] => [1,0,1,0] => 1
([(1,3),(2,3)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1
([(3,4)],5) => [1] => [1,0,1,0] => [1,0,1,0] => 1
([(2,4),(3,4)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(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] => [1,1,1,1,0,0,0,1,0,0] => 2
([(4,5)],6) => [1] => [1,0,1,0] => [1,0,1,0] => 1
([(3,5),(4,5)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(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] => [1,0,1,1,1,0,0,1,0,0] => 5
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(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] => [1,1,0,1,1,0,0,0,1,0] => 5
([(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] => [1,0,1,1,1,0,0,1,0,0] => 5
([(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] => [1,1,0,1,1,0,0,0,1,0] => 5
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(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] => [1,0,1,1,1,0,0,1,0,0] => 5
([(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] => [1,1,1,1,0,0,0,1,0,0] => 2
([(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] => [1,1,0,1,1,0,0,1,0,0] => 2
([(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] => [1,1,0,1,1,0,0,0,1,0] => 5
([(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] => [1,0,1,1,1,0,0,1,0,0] => 5
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(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] => [1,1,0,1,1,0,0,0,1,0] => 5
([(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] => [1,0,1,1,1,0,0,1,0,0] => 5
([(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] => [1,0,1,1,1,0,0,1,0,0] => 5
([(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] => [1,0,1,1,1,0,0,1,0,0] => 5
([(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] => [1,1,1,0,0,0,1,0,1,0] => 1
([(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] => [1,1,0,1,1,0,0,1,0,0] => 2
([(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] => [1,1,1,1,0,0,0,1,0,0] => 2
([(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] => [1,1,0,1,1,0,0,1,0,0] => 2
([(5,6)],7) => [1] => [1,0,1,0] => [1,0,1,0] => 1
([(4,6),(5,6)],7) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(3,6),(4,5)],7) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(3,6),(4,5),(5,6)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(2,3),(4,6),(5,6)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(4,5),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(1,2),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(3,6),(4,5),(4,6),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 5
([(3,5),(3,6),(4,5),(4,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1
([(1,6),(2,6),(3,5),(4,5)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => 5
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 5
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => 5
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => 1
([(1,6),(2,5),(3,4)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(2,6),(3,5),(4,5),(4,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(1,2),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(0,3),(1,2),(4,6),(5,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(2,3),(4,5),(4,6),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 5
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searching the database for the individual values of this statistic
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Map
switch returns and last double rise
Description
An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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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