Identifier
-
Mp00276:
Graphs
—to edge-partition of biconnected components⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤ
Values
([(0,2),(1,2)],3) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(1,3),(2,3)],4) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(0,3),(1,2)],4) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(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] => 1
([(2,4),(3,4)],5) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(1,4),(2,3)],5) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,4),(1,2),(1,3),(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] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(3,5),(4,5)],6) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
([(2,5),(3,4)],6) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
([(1,5),(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] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(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] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 5
([(4,6),(5,6)],7) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
([(3,6),(4,5)],7) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(3,6),(4,5),(5,6)],7) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(2,3),(4,6),(5,6)],7) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 2
([(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(1,2),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
([(3,6),(4,5),(4,6),(5,6)],7) => [3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
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Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
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
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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