Identifier
-
Mp00275:
Graphs
—to edge-partition of connected components⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001500: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(1,2)],3) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(0,2),(1,2)],3) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 2
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(2,3)],4) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(1,3),(2,3)],4) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 2
([(0,3),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(3,4)],5) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(2,4),(3,4)],5) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 2
([(1,4),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,4),(1,4),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
([(1,4),(2,3),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,1),(2,4),(3,4)],5) => [2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => 4
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,4),(1,4),(2,3),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(1,4),(2,3),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,4),(1,3),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(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] => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(4,5)],6) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(3,5),(4,5)],6) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 2
([(2,5),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(1,5),(2,5),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
([(2,5),(3,4),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(1,2),(3,5),(4,5)],6) => [2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => 4
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(1,5),(2,5),(3,4),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 4
([(2,5),(3,4),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(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) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 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] => 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 4
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(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] => [1,0,1,1,1,1,0,0,1,0,0,0] => 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(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] => [1,1,1,0,0,0,1,1,0,0] => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(5,6)],7) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(4,6),(5,6)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 2
([(3,6),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(2,6),(3,6),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 3
([(3,6),(4,5)],7) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
([(3,6),(4,5),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
>>> Load all 185 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The global dimension of magnitude 1 Nakayama algebras.
We use the code below to translate them to Dyck paths.
We use the code below to translate them to Dyck paths.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
Cori-Le Borgne involution
Description
The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Map
to edge-partition of connected components
Description
Sends a graph to the partition recording the number of edges in its connected components.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!