- FindStat

Identifier

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001198: Dyck paths ⟶ ℤ (values match St001206The maximal dimension of an indecomposable projective $eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$ .)

Values

([(0,1),(0,2),(1,2)],3) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(0,3),(1,3),(2,3)],4) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(0,3),(1,2)],4) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2

([(0,3),(1,2),(2,3)],4) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(1,2),(1,3),(2,3)],4) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,4),(2,4),(3,4)],5) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(0,4),(1,4),(2,4),(3,4)],5) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(1,4),(2,3)],5) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2

([(1,4),(2,3),(3,4)],5) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(0,1),(2,4),(3,4)],5) => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 2

([(2,3),(2,4),(3,4)],5) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(0,4),(1,4),(2,3),(3,4)],5) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(1,4),(2,3),(2,4),(3,4)],5) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,4),(1,3),(2,3),(2,4)],5) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 2

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,5),(3,5),(4,5)],6) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(1,5),(2,5),(3,5),(4,5)],6) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(2,5),(3,4)],6) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2

([(2,5),(3,4),(4,5)],6) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(1,2),(3,5),(4,5)],6) => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 2

([(3,4),(3,5),(4,5)],6) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(1,5),(2,5),(3,4),(4,5)],6) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 2

([(2,5),(3,4),(3,5),(4,5)],6) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 2

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,5),(2,4),(3,4),(3,5)],6) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 2

([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 2

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2

([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 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] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3

([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2

([(3,6),(4,6),(5,6)],7) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(2,6),(3,6),(4,6),(5,6)],7) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(3,6),(4,5)],7) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2

([(3,6),(4,5),(5,6)],7) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(2,3),(4,6),(5,6)],7) => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 2

([(4,5),(4,6),(5,6)],7) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2

([(2,6),(3,6),(4,5),(5,6)],7) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(1,2),(3,6),(4,6),(5,6)],7) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 2

([(3,6),(4,5),(4,6),(5,6)],7) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(3,5),(3,6),(4,5),(4,6)],7) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(1,6),(2,6),(3,5),(4,5)],7) => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 2

([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

>>> Load all 163 entries. <<<

([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2

([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,6),(3,5),(4,5),(4,6)],7) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3

([(1,2),(3,6),(4,5),(5,6)],7) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 2

([(0,3),(1,2),(4,6),(5,6)],7) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 2

([(2,3),(4,5),(4,6),(5,6)],7) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 2

([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3

([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,6),(2,5),(3,4),(3,5),(4,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3

([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2

([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2

([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3

([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 2

([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3

([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3

([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2

([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 2

([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 3

([(0,1),(2,5),(3,4),(4,6),(5,6)],7) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3

([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3

([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3

([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3

([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2

([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2

([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3

([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3

([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 3

search for individual values

searching the database for the individual values of this statistic

Description

The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

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.

Map

to edge-partition of connected components

Description

Sends a graph to the partition recording the number of edges in its connected components.

Map

Delest-Viennot

Description

Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
The Delest-Viennot bijection

$\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path

$(\gamma^{(-1)}\circ\beta)(D)$ .